a .wo.H ' OME UNIVERSITY LIBRARY F MODERN KNOWLEDGE AN INTRODUCTION TO MATHEMATICS BY A. N. WHITEHEAD, Se.D., F.R.S. LONDON WILLIAMS & NORGATE HKNRY HOLT & Co., NEW YORK CANADA: WM. BRIGGS, TORONTO INDIA : R. & T. WASHBOURNE, LTD. HOME UNIVERSITY LIBRARY OF MODERN KNOWLEDGE Editor* t HERBERT FISHER, M.A.. F.B.A. PROF. GILBERT MURRAY, D.LlTT., LL.D., F.B.A. PROF. J. ARTHUR THOMSON, M.A. PROF. WILLIAM T. BKEWSTER, M.A. COLUMBIA. UNIVERSITY, U.S.A.) NEW YORK HENRY HOLT AND COMPANY , iifr A /uilfJ I AN INTRODUCTION TO MATHEMATICS BY A. N. WHITEHEAD, SC.D- F.R.S., AUTHOR OF "UNIVERSAL ALGEBRA," JOINT AUTHOR OF " I'RINCIPIA MATHBHATICA " NEW AND REVISED EDITION IRV^T 1 LONDON WILLIAMS AND NORGATE PRINTED BT HAKELJy, WATSON AND VINET, LONDON AND AYLBSBUBT. CONTENTS OH&P. PASH I THE ABSTRACT NATURE OF MATHE- MATICS ..... 7 H VARIABLES . . , . . 15 m METHODS OF APPLICATION . . 25 IV DYNAMICS ..... 42 V THE SYMBOLISM OF MATHEMATICS . 58 VI GENERALIZATIONS OF NUMBER . . 71 VH IMAGINARY NUMBERS ... 87 VIH IMAGINARY NUMBERS (CONTINUED) . 101 IX COORDINATE GEOMETRY . . .112 X CONIC SECTIONS . . . .128 XI FUNCTIONS . . . . . 145 XII PERIODICITY IN NATURE . . .164 v vi CONTENTS CHAP. PAGE TRIGONOMETRY . . . .173 XIV SERIES .,..., 194 XV THE DIFFERENTIAL CALCULUS . . 217 XVI GEOMETRY . . . . 236 XVH QUANTITY . . . . . 245 NOTES ...... 250 BIBLIOGRAPHY . . .251 INDEX 253 AN INTEODUCTION TO MATHEMATICS CHAPTER I THE ABSTRACT NATURE OF MATHEMATICS THE study of mathematics is apt to com' mence in disappointment. The important applications of the science, the theoretical interest of its ideas, and the logical rigour of its methods, all generate the expectation of a speedy introduction to processes of interest. We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet's father, this great science eludes the efforts of our mental weapons to grasp it " 'Tis here, 'tis there, 'tis gone " and what we do see does not suggest the same excuse for illusiveness as sufficed for the ghost, that it is too noble for our gross methods. '* A show of violence," if ever excusable, may surely be " offered " to the trivial results which occupy the 7 8 INTRODUCTION TO MATHEMATICS pages of some elementary mathematical treatises. The reason for this failure of the science to live up to its reputation is that its funda- mental ideas are not explained to the student disentangled from the technical procedure which has been invented to facilitate their exact presentation in particular instances. Accordingly, the unfortunate learner finds himself struggling to acquire a knowledge of a mass of details which are not illuminated by any general conception. Without a doubt, technical facility is a first requisite for valu- able mental activity : we shall fail to appre- ciate the rhythm of Milton, or the passion of Shelley, so long as we find it necessary to spell the words and are not quite certain of the forms of the individual letters. In this sense there is no royal road to learning. But it is equally an error to confine attention to technical processes, excluding consideration of general ideas. Here lies the road to pedantry. The object of the following Chapters is not to teach mathematics, but to enable students from the very beginning of their course to know what the science is about, and why it is necessarily the foundation of exact thought as applied to natural phenomena. All allu- sion in what follows to detailed deductions in any part of the science will be inserted NATURE OF MATHEMATICS 9 merely for the purpose of example, and care will be taken to make the general argument comprehensible, even if here and there some technical process or symbol which the reader does not understand is cited for the purpose of illustration. The first acquaintance which most people have with mathematics is through arithmetic. That two and two make four is usually taken as the type of a simple mathematical pro- position which everyone will have heard of. Arithmetic, therefore, will be a good subject to consider in order to discover, if possible, the most obvious characteristic of the science. Now, the first noticeable fact about arithmetic is that it applies to everything, to tastes and to sounds, to apples and to angels, to the ideas of the mind and to the bones of the body. The nature of the things is perfectly indifferent, of all things it is true that two and two make four. Thus we write down as the leading characteristic of mathematics that it deals with properties and ideas which are applicable to things just because they are things, and apart from any particular feelings, or emotions, or sensations, in any way connected with them. This is what is meant by calling mathematics an abstract science. The result which we have reached deserves attention. It is natural to think that an 10 INTRODUCTION TO MATHEMATICS abstract science cannot be of much import- ance in the affairs of human life, because it has omitted from its consideration every- thing of real interest. It will be remembered that Swift, in his description of Gulliver's voyage to Laputa, is of two minds on this point. He describes the mathematicians of that country as silly and useless dreamers, whose attention has to be awakened by flappers. Also, the mathematical tailor mea- sures his height by a quadrant, and deduces his other dimensions by a rule and compasses, producing a suit of very ill-fitting clothes. On the other hand, the mathematicians of Laputa, by their marvellous invention of the magnetic island floating in the air, ruled the country and maintained their ascendency over their subjects. Swift, indeed, lived at a time peculiarly unsuited for gibes at con- temporary mathematicians. Newton's Prin- cipia had just been written, one of the great forces which have transformed the modern world. Swift might just as well have laughed at an earthquake. But a mere list of the achievements of mathematics is an unsatisfactory way of arriving at an idea of its importance. It is worth while to spend a little thought in getting at the root reason why mathematics, because of its very abstractness, must always remain one of the most important topics NATURE OF MATHEMATICS 11 for thought. Let us try to make clear to ourselves why explanations of the order of events necessarily tend to become mathe- matical. Consider how all events are interconnected. When we see the lightning, we listen for the thunder ; when we hear the wind, we look for the waves on the sea ; in the chill autumn, the leaves fall. Everywhere order reigns, so that when some circumstances have been noted we can foresee that others will also be present. The progress of science consists in observing these interconnections and in show- ing with a patient ingenuity that the events of this evershifting world are but examples of a few general connections or relations called laws. To see what is general in what is par- ticular and what is permanent in what is transitory is the aim of scientific thought. In the eye of science, the fall of an apple, the motion of a planet round a sun, and the cling- ing of the atmosphere to the earth are all seen as examples of the law of gravity. This possibility of disentangling the most complex evanescent circumstances into various ex- amples of permanent laws is the controlling idea of modern thought. Now let us think of the sort of laws which we want in order completely to realize this scientific ideal. Our knowledge of the par- ticular facts of the world around us is gained 12 INTRODUCTION TO MATHEMATICS from our sensations. We see, and hear, and taste, and smell, and feel hot and cold, and push, and rub, and ache, and tingle. These are just our own personal sensations : my toothache cannot be your toothache, and my sight cannot be your sight. But we ascribe the origin of these sensations to relations be- tween the things which form the external world. Thus the dentist extracts not the toothache but the tooth. And not only so, we also endeavour to imagine the world as one connected set of things which underlies all the perceptions of all people. There is not one world of things for my sensations and an- other for yours, but one world in which we both exist. It is the same tooth both for dentist and patient. Also we hear and we touch the same world as we see. It is easy, therefore, to understand that we want to describe the connections between these external things in some way which does not depend on any particular sensations, nor even on all the sensations of any particular person. The laws satisfied by the course of events in the world of external things are to be described, if possible, in a neutral uni- versal fashion, the same for blind men as for deaf men, and the same for beings with faculties beyond our ken as for normal human beings. But when we have put aside our immediate NATURE OF MATHEMATICS 13 sensations, the most serviceable part from its clearness, definiteness, and universality of what is left is composed of our general ideas of the abstract formal properties of things; in fact, the abstract mathematical ideas men- tioned above. Thus it comes about that, step by step, and not realizing the full mean- ing of the process, mankind has been led to search for a mathematical description of the properties of the universe, because in this way only can a general idea of the course of events be formed, freed from reference to particular persons or to particular types of sensation. For example, it might be asked at dinner: " What was it which underlay my sensation of sight, yours of touch, and his of taste and smell ? " the answer being " an apple." But in its final analysis, science seeks to describe an apple in terms of the positions and motions of molecules, a description which ignores me and you and him, and also ig- nores sight and touch and taste and smell. Thus mathematical ideas, because they are abstract, supply just what is wanted jfor a scientific description of the course 01 events. This point has usually been misunderstood, from being thought of in too narrow a way. Pythagoras had a glimpse of it when he pro- claimed that number was the source of all things. In modern times the belief that the 14 INTRODUCTION TO MATHEMATICS ultimate explanation of all things was to be found in Newtonian mechanics was an adum- bration of the truth that all science as it grows towards perfection becomes mathe- matical in its ideas. CHAPTER II VARIABLES MATHEMATICS as a science commenced when first someone, probably a Greek, proved pro- positions about any things or about some things, without specification of definite par- ticular things. These propositions were first enunciated by the Greeks for geometry ; and, accordingly, geometry was the great Greek mathematical science. After the rise of geo- metry centuries passed away before algebra made a really effective start, despite some faint anticipations by the later Greek mathe- maticians. The ideas of any and of some are intro- duced into algebra by the use of letters, in- stead of the definite numbers of arithmetic. Thus, instead of saying that 2+3=3+2, in algebra we generalize and say that, if x and y stand for any two numbers, then x -\-y =y +a?. Again, in the place of saying that 3 > 2, we generalize and say that if x be any number there exists some number (or numbers) y such that y>x. We may remark in passing that this latter assumption for when put in its strict ultimate form it is an assumption is 15 16 INTRODUCTION TO MATHEMATICS of vital importance, both to philosophy and to mathematics ; for by it the notion of in- finity is introduced. Perhaps it required the introduction of the arabic numerals, by which the use of letters as standing for definite numbers has been completely discarded in mathematics, in order to suggest to mathe- maticians the technical convenience of the use of letters for the ideas of any number and some number. The Romans would have stated the number of the year in which this is written in the form MDCCCCX., whereas we write it 1910, thus leaving the letters for the other usage. But this is merely a specu- lation. After the rise of algebra the differ- ential calculus was invented by Newton and Leibniz, and then a pause in the progress of the philosophy of mathematical thought occurred so far as these notions are concerned ; and it was not till within the last few years that it has been realized how fundamental any and some are to the very nature of mathe- matics, with the result of opening out still further subjects for mathematical explora- tion. Let us now make some simple algebraic statements, with the object of understanding exactly how these fundamental ideas occur. (1) For any number a?, #+2=2+#; (2) For some number #, #+2=8 ; (3) For some number a?, x +2 > 3. VARIABLES 17 The first point to notice is the possibilities contained in the meaning of some, as here used. Since a +2 =2 +x for any number a?, it is true for some number x. Thus, as here used, any implies some and some does not exclude any. Again, in the second example, there is, in fact, only one number x, such that x +2 =8, namely only the number 1. Thus the some may be one number only. But in the third, example, any number x which is greater than 1 gives x -f 2 > 3. Hence there are an infinite number of numbers which answer to the some number in this case. Thus some may be any- thing between any and one only, including both these limiting cases. It is natural to supersede the statements (2) and (3) by the questions : (2') For what number x is x +2 =3; (3') For what numbers x is #-f2>3. Considering (2'), #+2=3 is an equation, and it is easy to see that its solution is x =3 2 =1. When we have asked the question implied in the statement of the equation <r+2=8, x is called the unknown. The object of the solu- tion of the equation is the determination of the unknown. Equations are of great im- portance in mathematics, and it seems as though (2') exemplified a much more thorough- going and fundamental idea than the original statement (2). This, however, is a complete mistake. The idea of the undetermined 18 INTRODUCTION TO MATHEMATICS " variable " as occurring in the use of ** some " or " any " is the really important one in mathematics ; that of the " unknown " in an equation, which is to be solved as quickly as possible, is only of subordinate use, though of course it is very important. One of the causes of the apparent triviality of much of elementary algebra is the preoccupation of the text-books with the solution of equations. The same remark applies to the solution of the inequality (3') as compared to the original statement (3). But the majority of interesting formulae, especially when the idea of some is present, involve more than one variable. For ex- ample, the consideration of the pairs of num- bers x and y (fractional or integral) which satisfy x+y=I involves the idea of two corre- lated variables, x and y. When two variables are present the same two main types of statement occur. For example, (1) for any pair of numbers, x and y, x+y=y+ac, and (2) for some pairs of numbers, x and t/, The second type of statement invites con- sideration of the aggregate of pairs of num- bers which are bound together by some fixed relation in the case given, by the relation x+y=\. One use of formulae of the first type, true for any pair of numbers, is that by them formulae of the second type can be VARIABLES 19 thrown into an indefinite number of equiva- lent forms. For example, the relation x-\-y =1 is equivalent to the relations y+x=I, (x-y)+2y=l, 6x+6y=6, and so on. Thus a skilful mathematician uses that equivalent form of the relation under consideration which is most convenient for his immediate purpose. It is not in general true that, when a pair of terms satisfy some fixed relation, if one of the terms is given the other is also definitely determined. For example, when x and y satisfy y 2 =x, if #=4, y can be 2, thus, for any positive value of x there are alter- native values for y. Also in the relation j?+t/>l, when either x or y is given, an indefinite number of values remain open for the other. Again there is another important point to be noticed. If we restrict ourselves to posi- tive numbers, integral or fractional, in con- sidering the relation <c+t/=l, then, if either x or y be greater than 1, there is no positive number which the other can assume so as to satisfy the relation. Thus the "field" of the relation for x is restricted to numbers less than 1, and similarly for the " field " open to y. Again, consider integral numbers only, positive or negative, and take the relation 20 INTRODUCTION TO MATHEMATICS t/ 2 =#, satisfied by pairs of such numbers. Then whatever integral value is given to y, x can assume one corresponding integral value. So the " field " for y is unrestricted among these positive or negative integers. But the " field " for x is restricted in two ways. In the first place x must be positive, and in the second place, since y is to be in- tegral, x must be a perfect square. Accord- ingly, the " field " of x is restricted to the set of integers I 2 , 2 2 , 3 2 , 4 2 , and so on, i.e., to 1, 4, 9, 16, and so on. The study of the general properties of a relation between pairs of numbers is much facilitated by the use of a diagram constructed as follows : O x M i A Fig. 1. Draw two lines OX and OF at right angles ; let any number x be represented by x units VARIABLES 21 (in any scale) of length along OX, any num- ber ybyy units (in any scale) of length along OF. Thus if OM, along OX, be x units in length, and ON, along OF, be y units in length, by completing the parallelogram OMPN we find a point P which corresponds to the pair of numbers x and y. To each point there corresponds one pair of numbers, and to each pair of numbers there corresponds one point. The pair of numbers are called the co- ordinates of the point. Then the points whose coordinates satisfy some fixed rela- tion can be indicated in a convenient way, by drawing a line, if they all lie on a line, or by shading an area if they are all points in the area. If the relation can be repre- sented by an equation such as o?+t/=l, or t/ 2 =#, then the points lie on a line, which is straight in the former case and curved in the latter. For example, considering only positive numbers, the points whose co- ordinates satisfy x-\-y=\ lie on the straight line AB in Fig. 1, where 0^=1 and OB=l. Thus this segment of the straight line AB gives a pictorial representation of the proper- ties of the relation under the restriction to positive numbers. Another example of a relation between two variables is afforded by considering the varia- tions in the pressure and volume of a given mass of some gaseous substance such as air 22 INTRODUCTION TO MATHEMATICS or coal-gas or steam at a constant tempera- ture. Let v be the number of cubic feet in its volume and p its pressure in Ib. weight per square inch. Then the law, known as Boyle's law, expressing the relation between p and v as both vary, is that the product pv is constant, always supposing that the temperature does not alter. Let us suppose, for example, that the quantity of the gas and its other circumstances are such that we can put pv=I (the exact number on the right-hand side of the equation makes no essential difference). Then in Fig. 2 we take two lines, OV and OP, at right angles and draw OM along OV to represent v units of volume, and ON along VARIABLES 23 OP to represent p units of pressure. Then the point Q, which is found by completing the parallelogram OMQN, represents the state of the gas when its volume is v cubic feet and its pressure is p Ib. weight per square inch. If the circumstances of the portion of gas con- sidered are such that pv=l, then all these points Q which correspond to any possible state of this portion of gas must lie on the curved line ABC, which includes all points for which p and v are positive, and pv=I. Thus this curved line gives a pictorial repre- sentation of the relation holding between the volume and the pressure. When the pressure is very big the corresponding point Q must be near C, or even beyond C on the undrawn part of the curve ; then the volume will be very small. When the volume is big Q will be near to A, or beyond A ; and then the pressure will be small. Notice that an en- gineer or a physicist may want to know the particular pressure corresponding to some definitely assigned volume. Then we have the case of determining the unknown p when v is a known number. But this is only in particular cases. In considering generally the properties of the gas and how it will be- have, he has to have in his mind the general form of the whole curve ABC and its general properties. In other words the really funda- mental idea is that of the pair of variables 24 INTRODUCTION TO MATHEMATICS satisfying the relation pv=I. This example illustrates how the idea of variables is funda- mental, both in the applications as well as in the theory of mathematics. CHAPTER III METHODS OF APPLICATION THE way in which the idea of variables satisfying a relation occurs in the applications of mathematics is worth thought, and by devoting some time to it we shall clear up our thoughts on the whole subject. Let us start with the simplest of examples : Suppose that building costs Is. per cubic foot and that 205. make l. Then in all the complex circumstances which attend the building of a new house, amid all the various sensations and emotions of the owner, the architect, the builder, the workmen, and the onlookers as the house has grown to comple- tion, this fixed correlation is by the law assumed to hold between the cubic content and the cost to the owner, namely that if x be the number of cubic feet, and y the cost, then 20?/=a?. This correlation of x and y is assumed to be true for the building of any house by any owner. Also, the volume of the house and the cost are not supposed to have been perceived or apprehended by any particular sensation or faculty, or by any 25 26 INTRODUCTION TO MATHEMATICS particular man. They are stated in an ab- stract general way, with complete indiffer- ence to the owner's state of mind when he has to pay the bill. Now think a bit further as to what all this means. The building of a house is a com- plicated set of circumstances. It is im- possible to begin to apply the law, or to test it, unless amid the general course of events it is possible to recognize a definite set of occurrences as forming a particular instance of the building of a house. In short, we must know a house when we see it, and must recog- nize the events which belong to its building. Then amidst these events, thus isolated in idea from the rest of nature, the two elements of the cost and cubic content must be deter- minable ; and when they are both determined, if the law be true, they satisfy the general formula 20y=a. *But is tl ! ( law true ? Anyone who has had much to i ; >. with building will know that we have hext put the cost rather high. It is only for <>a expensive type of house that it will work out at this price. This brings out another point which must be made clear. While we are making mathematical calcula- tions connected with the formula 20t/=#, it is indifferent to us whether the law be true or METHODS OF APPLICATION 27 false. In fact, the very meanings assigned to x and y, as being a number of cubic feet and a number of pounds sterling, are in- different. During the mathematical investi- gation we are, in fact, merely considering the properties of this correlation between a pair of variable numbers x and y. OUT results will apply equally well, if we interpret y to mean a number of fishermen and x the num- ber of fish caught, so that the assumed law is that on the average each fisherman catches twenty fish. The mathematical certainty of the investigation only attaches to the results considered as giving properties of the corre- lation 20y=x between the variable pair of numbers x and y. There is no mathematical certainty whatever about the cost of the actual building of any house. The law is not quite true and the result it gives will not be quite accurate. In fact, it may well be hope- lessly wrong. Now all this no doubt seems very obvious. But in truth with more complicated instances there is no more common error than to assume that, because prolonged and accurate mathe- matical calculations have been made, the application of the result to some fact of nature is absolutely certain. The conclusion of no argument can be more certain than the assumptions from which it starts. All mathe- matical calculations about the course of 28 INTRODUCTION TO MATHEMATICS nature must start from some assumed law of nature, such, for instance, as the assumed law of the cost of building stated above. Accordingly, however accurately we have calculated that some event must occur, the doubt always remains Is the law true ? If the law states a precise result, almost cer- tainly it is not precisely accurate ; and thus even at the best the result, precisely as calcu- lated, is not likely to occur. But then we have no faculty capable of observation with ideal precision, so, after all, our inaccurate laws may be good enough. We will now turn to an actual case, that of Newton and the Law of Gravity. This law states that any two bodies attract one an- other with a force proportional to the product of their masses, and inversely proportional to the square of the distance between them. Thus if m and M are the masses of the two bodies, reckoned in Ibs. say, and d miles is the distance between them, the force on either body, due to the attraction of the other and directed towards it, is proportional to p- ; thus this force can be written as equal to JT~ wnere & is a definite number depending on the absolute magnitude of this attraction and also on the scale by which we choose to measure forces. It is easy to see that, if we METHODS OF APPLICATION 29 wish to reckon in terms of forces such as the weight of a mass of 1 lb., the number which k represents must be extremely small ; for when m and M and d are each put equal to 1, becomes the gravitational attraction of two equal masses of 1 lb. at the distance of one mile, and this is quite inappreciable. However, we have now got our formula for the force of attraction. If we call this force F, it is F^kp-, giving the correlation be- tween the variables F, m, M, and d. We all know the story of how it was found out. Newton, it states^ was sitting in an orchard and watched the fall of an apple, and then the law of universal gravitation burst upon his mind. It may be that the final formu- lation of the law occurred to him in an orchard, as well as elsewhere and he must have been somewhere. But for our purposes it is more instructive to dwell upon the vast amount of preparatory thought, the product of many minds and many centuries, which was necessary before this exact law could be formulated. In the first place, the mathe- matical habit of mind and the mathematicaJ procedure explained in the previous two chapters had to be generated ; otherwise Newton could never have thought of a formula representing the force between any two masses 30 INTRODUCTION TO MATHEMATICS at any distance. Again, what are the mean- ings of the terms employed, Force, Mass, Dis- tance ? Take the easiest of these terms, Distance. It seems very obvious to us to conceive all material things as forming a de- finite geometrical whole, such that the dis- tances of the various parts are measurable in terms of some unit length, such as a mile or a yard. This is almost the first aspect of a material structure which occurs to us. It is the gradual outcome of the study of geometry and of the theory of measurement. Even now, in certain cases, other modes of thought are convenient. In a mountainous country distances are often reckoned in hours. But leaving distance, the other terms, Force and Mass, are much more obscure. The exact comprehension of the ideas which Newton meant to convey by these words was of slow growth, and, indeed, Newton himself was the first man who had thoroughly mastered the true general principles of Dynamics. Throughout the middle ages, under the in- fluence of Aristotle, the science was entirely misconceived. Newton had the advantage of coming after a series of great men, notably Galileo, in Italy, who in the previous two centuries had reconstructed the science and had invented the right way of thinking about it. He completed their work. Then, finally, having the ideas of force, mass, and distance, clear and distinct in his mind, and realising their importance and their relevance to the fall of an apple and the motions of the planets, he hit upon the law of gravitation and proved it to be the formula always satisfied in these various motions. The vital point in the application of mathe- matical formulae is to have clear ideas and a correct estimate of their relevance to the phenomena under observation. No less than ourselves, our remote ancestors were im- pressed with the importance of natural phenomena and with the desirability of taking energetic measures to regulate the sequence of events. Under the influence of irrelevant ideas they executed elaborate religious cere- monies to aid the birth of the new moon, and performed sacrifices to save the sun during the crisis of an eclipse. There is no reason to believe that they were more stupid than we are. But at that epoch there had not been opportunity for the slow accumulation of clear and relevant ideas. The sort of way in which physical sciences grow into a form capable of treatment by mathematical methods is illustrated by the history of the gradual growth of the science of electromagnetism. Thunderstorms are events on a grand scale, arousing terror in men and even animals. From the earliest times they must have been objects of wild 32 INTRODUCTION TO MATHEMATICS and fantastic hypotheses, though it may be doubted whether our modern scientific dis- coveries in connection with electricity are not more astonishing than any of the magical explanations of savages. The Greeks knew that amber (Greek, electron) when rubbed would attract light and dry bodies. In 1600 A.D., Dr. Gilbert, of Colchester, published the first work on the subject in which any scientific method is followed. He made a list of substances possessing properties similar to those of amber ; he must also have the credit of connecting, however vaguely, electric and magnetic phenomena. At the end of the seventeenth and throughout the eighteenth century knowledge advanced. Electrical machines were made, sparks were obtained from them ; and the Leyden Jar was in- vented, by which these effects could be in- tensified. Some organised knowledge was being obtained ; but still no relevant mathe- matical ideas had been found out. Franklin, in the year 1752, sent a kite into the clouds and proved that thunderstorms were elec- trical. Meanwhile from the earliest epoch (2634 B.C.) the Chinese had utilized the characteristic property of the compass needle, but do not seem to have connected it with any theoretical ideas. The really profound changes in human life all have their ultimate origin in knowledge METHODS OF APPLICATION 33 pursued for its own sake. The use of the com- pass was not introduced into Europe till the end of the twelfth century A.D., more than 3000 years after its first use in China. The import- ance which the science of electromagnetism has since assumed in every department of human life is not due to the superior practical bias of Europeans, but to the fact that in the West electrical and magnetic phenomena were studied by men who were dominated by abstract theoretic interests. The discovery of the electric current is due to two Italians, Galvani in 1780, and Volta in 1792. This great invention opened a new series of phenomena for investigation. The scientific world had now three separate, though allied, groups of occurrences on hand the effects of " statical " electricity arising from frictional electrical machines, the mag- netic phenomena, and the effects due to electric currents. From the end of the eighteenth century onwards, these three lines of investigation were quickly inter-connected and the modern science of electromagnetism was constructed, which now threatens to transform human life. Mathematical ideas now appear. During the decade 1780 to 1789, Coulomb, a French- man, proved that magnetic poles attract or repel each other, in proportion to the inverse square of their distances, and also that the 34 INTRODUCTION TO MATHEMATICS same law holds for electric charges laws curiously analogous to that of gravitation. In 1820, Oersted, a Dane, discovered that electric currents exert a force on magnets, and almost immediately afterwards the mathematical law of the force was correctly formulated by Ampere, a Frenchman, who also proved that two electric currents exerted forces on each other. " The experimental in- vestigation by which Ampere established the law of the mechanical action between electric currents is one of the most brilliant achieve- ments in science. The whole, theory and experiment, seems as if it had leaped, full- grown and full armed, from the brain of the ' Newton of Electricity.' It is perfect in form, and unassailable in accuracy, and it is summed up in a formula from which all the phenomena may be deduced, and which must always remain the cardinal formula of electro-dynamics." * The momentous laws of induction between currents and between currents and magnets were discovered by Michael Faraday in 1831- 82. Faraday was asked: "What is the use of this discovery ? " He answered : " What is the use of a child it grows to be a man." Faraday's child has grown to be a man and is now the basis of all the modern applications * Electricity and Magnetism, Clerk Maxwell, VoL II., eh. iii. METHODS OF APPLICATION 35 of electricity. Faraday also reorganized the whole theoretical conception of the science. His ideas, which had not been fully under- stood by the scientific world, were extended and put into a directly mathematical form by Clerk Maxwell in 1873. As a result of his mathematical investigations, Maxwell recog- nized that, under certain conditions, electrical vibrations ought to be propagated. He at once suggested that the vibrations which form light are electrical. This suggestion has since been verified, so that now the whole theory of light is nothing but a branch of the great science of electricity. Also Herz, a German, in 1888, following on Maxwell's ideas, succeeded in producing electric vibra- tions by direct electrical methods His experiments are the basis of our wireless telegraphy. In more recent years even more funda- mental discoveries have been made, and the science continues to grow in theoretic import- ance and in practical interest. This rapid sketch of its progress illustrates how, by the gradual introduction of the relevant theoretic ideas, suggested by experiment and them- selves suggesting fresh experiments, a whole mass of isolated and even trivial phenomena are welded together into one coherent science, in which the results of abstract mathematical deductions, starting from a few simple as- 36 INTRODUCTION TO MATHEMATICS sumed laws, supply the explanation to the complex tangle of the course of events. Finally, passing beyond the particular sciences of electromagnetism and light, we can generalize our point of view still further, and direct our attention to the growth of mathematical physics considered as one great chapter of scientific thought. In the first place, what in the barest outlines is the story of its growth ? It did not begin as one science, or as the product of one band of men. The Chaldean shepherds watched the skies, the agents of Government in Mesopotamia and Egypt measured the land, priests and philosophers brooded on the general nature of all things. The vast mass of the operations of nature appeared due to mysterious unfathomable forces. " The wind bloweth where it listeth ? ' expresses accurately the blank ignorance then existing of any stable rules followed in detail by the succession of phenomena. In broad out- line, then as now, a regularity of events was patent. But no minute tracing of their inter- connection was possible, and there was no knowledge how even to set about to construct such a science. Detached speculations, a few happy or un- happy shots at the nature of things, formed the utmost which could be produced. Meanwhile land-surveys had produced geo- METHODS OF APPLICATION 87 metry, and the observations of the heavens disclosed the exact regularity of the solar system. Some of the later Greeks, such as Archimedes, had just views on the elementary phenomena of hydrostatics and optics. In- deed, Archimedes, who combined a genius for mathematics with a physical insight, must rank with Newton, who lived nearly two thousand years later, as one of the founders of mathematical physics. He lived at Syra- cuse, the great Greek city of Sicily. When the Romans besieged the town (in 212 to 210 B.C.), he is said to have burned their ships by concentrating on them, by means of mirrors, the sun's rays. The story is highly improbable, but is good evidence of the repu- tation which he had gained among his con- temporaries for his knowledge of optics. At the end of this siege he was killed. According to one account given by Plutarch, in his life of Marcellus, he was found by a Roman soldier absorbed in the study of a geometrical diagram which he had traced on the sandy floor of his room. He did not immediately obey the orders of his captor, and so was killed. For the credit of the Roman generals it must be said that the soldiers had orders to spare him. The internal evidence for the other famous story of him is very strong ; for the discovery attributed to him is one eminently worthy of his genius for mathematical and physical re- 38 INTRODUCTION TO MATHEMATICS search. Luckily, it is simple enough to be explained here in detail. It is one of the best easy examples of the method of application of mathematical ideas to physics. Hiero, King of Syracuse, had sent a quan- tity of gold to some goldsmith to form the material of a crown. He suspected that the craftsmen had abstracted some of the gold and had supplied its place by alloying the remainder with some baser metal. Hiero sent the crown to Archimedes and asked him to test it. In these days an indefinite num- ber of chemical tests would be available. But then Archimedes had to think out the matter afresh. The solution flashed upon him as he lay in his bath. He jumped up and ran through the streets to the palace, shouting Eureka! Eureka! (I have found it, I have found it). This day, if we knew which it was, ought to be celebrated as the birthday of mathematical physics ; the science came of age when Newton sat in his orchard. Archimedes had in truth made a great discovery. He saw that a body when immersed in water is pressed upwards by the surrounding water with a resultant force equal to the weight of the water it displaces. This law can be proved theoretically from the mathematical principles of hydrostatics and can also be verified experimentally. Hence, if W Ib. be the weight of the crown, as weighed METHODS OF APPLICATION 39 in air, and w Ib. be the weight of the water which it displaces when completely immersed, W w would be the extra upward force necessary to sustain the crown as it hung in water. Now, this upward force can easily be ascer- tained by weighing the body as it hangs in water, as shown in the annexed figure. If The crown Weights Fig. 3. the weights in the right-hand scale come to F Ib., then the apparent weight of the crown in water is F Ib. ; and we thus have F=W-w and thus and W w W W-F where W and F are determined by the easy, and fairly precise, operation of weighing. 40 INTRODUCTION TO MATHEMATICS W Hence, by equation (A), is known. But W is the ratio of the weight of the crown to to the weight of an equal volume of water. This ratio is the same for any lump of metal of the same material : it is now called the specific gravity of the material, and depends only on the intrinsic nature of the substance and not on its shape or quantity. Thus to test if the crown were of gold, Archimedes had only to take a lump of indisputably pure gold and find its specific gravity by the same process. If the two specific gravities agreed, the crown was pure ; if they disagreed, it was debased. This argument has been given at length, because not only is it the first precise example of the application of mathematical ideas to physics, but also because it is a perfect and simple example of what must be the method and spirit of the science for all time. The death of Archimedes by the hands of a Roman soldier is symbolical of a world-change of the first magnitude : the theoretical Greeks, with their love of abstract science, were super- seded in the leadership of the European world by the practical Romans. Lord Beacons- field, in one of his novels, has defined a practi- cal man as a man who practises the errors of his forefathers. The Romans were a great race, but they were cursed with the sterility METHODS OF APPLICATION 41 which waits upon practicality. They did not improve upon the knowledge of their fore- fathers, and all their advances were confined to the minor technical details of engineering. They were not dreamers enough to arrive at new points of view, which could give a more fundamental control over the forces of nature. No Roman lost his life because he was ab- sorbed in the contemplation of a mathe- matical diagram. CHAPTER IV DYNAMICS THE world had to wait for eighteen hundred years till the Greek mathematical physicists found successors. In the sixteenth and seven- teenth centuries of our era great Italians, in particular Leonardo da Vinci, the artist (born 1452, died 1519), and Galileo (born 1564, died 1642), rediscovered the secret, known to Archimedes, of relating abstract mathematical ideas with the experimental investigation of natural phenomena. Meanwhile the slow advance of mathematics and the accumula- tion of accurate astronomical knowledge had placed natural philosophers in a much more advantageous position for research. Also the very egoistic self-assertion of that age, its greediness for personal experience, led its thinkers to want to see for themselves what happened ; and the secret of the relation of mathematical theory and experiment in in- ductive reasoning was practically discovered. It was an act eminently characteristic of the age that Galileo, a philosopher, should have 42 DYNAMICS 43 dropped the weights from the leaning tower of Pisa. There are always men of thought and men of action ; mathematical physics is the product of an age which combined in the same men impulses to thought with impulses to action. This matter of the dropping of weights from the tower marks picturesquely an essential step in knowledge, no less a step than the first attainment of correct ideas on the science of dynamics, the basal science of the whole subject. The particular point in dispute was as to whether bodies of different weights would fall from the same height in the same time. According to a dictum of Aristotle, universally followed up to that epoch, the heavier weight would fall the quicker. Gali- leo affirmed that they would fall in the same time, and proved his point by dropping weights from the top of the leaning tower. The apparent exceptions to the rule all arise when, for some reason, such as extreme light- ness or great speed, the air resistance is im- portant. But neglecting the air the law is exact. Galileo's successful experiment was not the result of a mere lu'cky guess. It arose from his correct ideas in connection with inertia and mass. The first law of motion, as follow- ing Newton we now enunciate it, is Every body continues in its state of rest or of uni- 44 INTRODUCTION TO MATHEMATICS form motion in a straight line, except so far as it is compelled by impressed force to change that state. This law is more than a dry formula : it is also a paean of triumph over defeated heretics. The point at issue can be understood by deleting from the law the phrase " or of uniform motion in a straight line." We there obtain what might be taken as the Aristotelian opposition formula: ** Every body continues in its state of rest except so far as it is compelled by impressed force to change that state." In this last false formula it is asserted that, apart from force, a body continues in a state of rest ; and accordingly that, if a body is moving, a force is required to sustain the motion ; so that when the force ceases, the motion ceases. The true Newtonian law takes diametrically the opposite point of view. The state of a body unacted on by force is that of uniform motion in & straight line, and no external force or influence is to be looked for as the cause, or, if you like to put it so, as the invariable accompaniment of this uniform rectilinear motion. Rest is merely a par- ticular case of such motion, merely when the velocity is and remains zero. Thus, when a body is moving, we do not seek for any ex- ternal influence except to explain changes in the rate of the velocity or changes in its direc- tion. So long as the body is moving at the DYNAMICS 45 same rate and in the same direction there is no need to invoke the aid of any forces. The difference between the two points of view is well seen by reference to the theory of the motion of the planets. Copernicus, a Pole, born at Thorn in West Prussia (born 1473, died 1543), showed how much simpler it was to conceive the planets, including the Force (on False hypothesis) Fig. 4. earth as revolving round the sun in orbits which are nearly circular ; and later, Kepler, a German mathematician, in the year 1609 proved that, in fact, the orbits are practically ellipses, that is, a special sort of oval curves which we will consider later in more detail. Immediately the question arose as to what are the forces which preserve the planets in this motion. According to the old false view, 46 INTRODUCTION TO MATHEMATICS held by Kepler, the actual velocity itself re- quired preservation by force. Thus he looked for tangential forces as in the accompanying figure (4). But according to the Newtonian law, apart from some force the planet would move for ever with its existing velocity in a straight line, and thus depart entirely from the sun. Newton, therefore, had to search for a force which would bend the motion Planer Fig. 5. round into its elliptical orbit. This he showed must be a force directed towards the sun as in the next figure (5). In fact, the force is the gravitational attraction of the sun acting according to the law of the inverse square of the distance, which has been stated above. The science of mechanics rose among the Greeks from a consideration of the theory of the mechanical advantage obtained by the use DYNAMICS 47 of a lever, and also from a consideration of various problems connected with the weights of bodies. It was finally put on its true basis at the end of the sixteenth and during the seventeenth centuries, as the preceding ac- count shows, partly with the view of explain- ing the theory of falling bodies, but chiefly in order to give a scientific theory of planetary motions. But since those days dynamics has taken upon itself a more ambitious task, and now claims to be the ultimate science of which the others are but branches. The claim amounts to this : namely, that the various qualities of things perceptible to the senses are merely our peculiar mode of appreciating changes in position on the part of things existing in space. For example, suppose we look at Westminster Abbey. It has been standing there, grey and immovable, for cen- turies past. But, according to modern scien- tific theory, that greyness, which so heightens our sense of the immobility of the building, is itself nothing but our way of appreciating the rapid motions of the ultimate molecules, which form the outer surface of the building and communicate vibrations to a substance called the ether. Again we lay our hands on its stones and note their cool, even temperature, so symbolic of the quiet repose of the building. But this feeling of temperature simply marks our sense of the transfer of heat from the 48 INTRODUCTION TO MATHEMATICS hand to the stone, or from the stone to the hand ; and, according to modern science, heat is nothing but the agitation of the mole- cules of a body. Finally, the organ begins playing, and again sound is nothing but the result of motions of the air striking on the drum of the ear. Thus the endeavour to give a dynamical explanation of phenomena is the attempt to explain them by statements of the general form, that such and such a substance or body was in this place and is now in that place. Thus we arrive at the great basal idea of modern science, that all our sensations are the result of comparisons of the changed configurations of things in space at various times. It follows therefore, that the laws of motion, that is, the laws of the changes of configurations of things, are the ultimate laws of physical science. In the application of mathematics to the investigation of natural philosophy, science does systematically what ordinary thought does casually. When we talk of a chair, we usually mean something which we have been seeing or feeling in some way ; though most of our language will presuppose that there is something which exists independently of our sight or feeling. Now in mathematical physics the opposite course is taken. The chair is conceived without any reference to DYNAMICS 49 anyone in particular, or to any special modes of perception. The result is that the chair becomes in thought a set of molecules in space, or a group of electrons, a portion of the ether in motion, or however the current scientific ideas describe it. But the point is that science reduces the chair to things moving in space and influencing each other's motions. Then the various elements or factors which enter into a set of circumstances, as thus conceived, are merely the things, like lengths of lines, sizes of angles, areas, and volumes, by which the positions of bodies in space can be settled. Of course, in addition to these geo- metrical elements the fact of motion and change necessitates the introduction of the rates of changes of such elements, that is to say, velocities, angular velocities, accelera- tions, and suchlike things. Accordingly, mathe- matical physics deals with correlations be- tween variable numbers which are supposed to represent the correlations which exist in nature between the measures of these geo- metrical elements and of their rates of change. But always the mathematical laws deal with variables, and it is only in the occasional testing of the laws by reference to experi- ments, or in the use of the laws for special predictions that definite numbers are substi- tuted. The interesting point about the world as 50 INTRODUCTION TO MATHEMATICS thus conceived in this abstract way through- out the study of mathematical physics, where only the positions and shapes of things are considered together with their changes, is that the events of such an abstract world are suffi- cient to "explain" our sensations. When we hear a sound, the molecules of the air have been agitated in a certain way : given the agitation, or air- waves as they are called, all normal people hear sound ; and if there are no air-waves, there is no sound. And, simi- larly, a physical cause or origin, or parallel event (according as different people might like to phrase it) underlies our other sensations. Our very thoughts appear to cprrespond to conformations and motions of the brain ; in- jure the brain and you injure the thoughts. Meanwhile the events of this physical universe succeed each other according to the mathe- matical laws which ignore all special sensa- tions and thoughts and emotions. Now, undoubtedly, this is the general aspect of the relation of the world of mathematical physics to our emotions, sensations, and thoughts ; and a great deal of controversy has been occasioned by it and much ink spilled. We need only make one remark. The whole situation has arisen, as we have seen, from the endeavour to describe an external world " explanatory " of our various in- dividual sensations and emotions, but a world DYNAMICS 51 also, not essentially dependent upon any particular sensations or upon any particular individual. Is such a world merely but one huge fairy tale ? But fairy tales are fantastic and arbitrary : if in truth there be such a world, it ought to submit itself to an exact description, which determines accurately its various parts and their mutual relations. Now, to a large degree, this scientific world does submit itself to this test and allow its events to be explored and predicted by the apparatus of abstract mathematical ideas. It certainly seems that here we have an inductive verification of our initial assumption. It must be admitted that no inductive proof is conclusive ; but if the whole idea of a world which has existence independently of our particular per- ceptions of it be erroneous, it requires careful explanation why the attempt to characterise it, in terms of that mathematical remnant of our ideas which would apply to it, should issue in such a remarkable success. It would take us too far afield to enter into a detailed explanation of the other laws of motion. The remainder of this chapter must be devoted to the explanation of remarkable ideas which are fundamental, both to mathe- matical physics and to pure mathematics : these are the ideas of vector quantities and the parallelogram law for vector addition. We 52 INTRODUCTION TO MATHEMATICS have seen that the essence of motion is that a body was at A and is now at C. This trans- ference from A to C requires two distinct elements to be settled before it is completely determined, namely its magnitude (i.e. the length AC) and its direction. Now any- thing, like this transference, which is com- pletely given by the determination of a magni- tude and a direction is called a vector. For example, a velocity requires for its definition the assignment of a magnitude and of a direction. It must be of so many miles per hour in such and such a direction. The ex- istence and the independence of these two elements in the determination of a velocity are well illustrated by the action of the captain of a ship, who communicates with different sub- ordinates respecting them : he tells the chief engineer the number of knots at which he is to steam, and the helmsman the compass DYNAMICS 58 bearing of the course which he is to keep. Again the rate of change of velocity, that is velocity added per unit time, is also a vector quantity : it is called the acceleration. Simi- larly a force in the dynamical sense is another vector quantity. Indeed, the vector nature of forces follows at once according to dynami- cal principles from that of velocities and accelerations ; but this is a point which we need not go into. It is sufficient here to say that a force acts on a body with a certain magnitude in a certain direction. Now all vectors can be graphically repre- sented by straight lines. All that has to be done is to arrange : (i) a scale according to which units of length correspond to units of magnitude of the vector for example, one inch to a velocity of 10 miles per hour in the case of velocities, and one inch to a force of 10 tons weight in the case of forces and (ii) a direction of the line on the diagram corre- sponding to the direction of the vector. Then a line drawn with the proper number of inches of length in the proper direction represents the required vector on the arbitrarily assigned scale of magnitude. This diagrammatic representa- tion of vectors is of the first importance. By its aid we can enunciate the famous " parallelo- gram law " for the addition of vectors of the same kind but in different directions. Consider the vector AC in figure 6 as repre- 54 INTRODUCTION TO MATHEMATICS sentative of the changed position of a body from A to C : we will call this the vector of transportation. It will be noted that, if the reduction of physical phenomena to mere changes in positions, as explained above, is correct, all other types of physical vectors are really reducible in some way or other to this single type. Now the final transportation from A to C is equally well effected by a transportation from A to B and a transporta- tion from B to C, or, completing the parallelo- gram ABCD, by a transportation from A to D and a transportation from D to C. These transportations as thus successively applied are said to be ad<ded together. This is simply a definition of what we mean by the addition of transportations. Note further that, con- sidering parallel lines as being lines drawn in the same direction, the transportations B to C and A to D may be conceived as the same transportation applied to bodies in the two initial positions B and A. With this con- ception we may talk of the transportation A to D as applied to a body in any position, for example at B. Thus we may say that the transportation A to C can be conceived as the sum of the two transportations A to B and A to D applied in any order. Here we have the parallelogram law for the ad- dition of transportations : namely, if the transportations are A to B and A to D, DYNAMICS 55 complete the parallelogram ABCD, and then the sum of the two is the diagonal AC. All this at first sight may seem to be very artificial. But it must be observed that nature itself presents us with the idea. For example, a steamer is moving in the direction AD (cf. fig. 6) and a man walks across its deck. If the steamer were still, in one minute he would arrive at B ; but during that minute his starting point A on the deck has moved to D, and his path on the deck has moved from AB to DC. So that, in fact, his transportation has been from A to C over the surface of the sea. It is, however, presented to us analysed into the sum of two transportations, namely, one from A to B relatively to the steamer, and one from A to D which is the transportation of the steamer. By taking into account the element of time, namely one minute, this diagram of the man's transportation AC represents his velocity. For if AC represented so many feet of trans- portation, it now represents a transportation of so many feet per minute, that is to say, it represents the velocity of the man. Then AB and AD represent two velocities, namely, his velocity relatively to the steamer, and the velocity of the steamer, whose "sum" makes up his complete velocity. It is evident that diagrams and definitions concerning trans- 56 INTRODUCTION TO MATHEMATICS portations are turned into diagrams and de- finitions concerning velocities by conceiving the diagrams as representing transportations per unit time. Again, diagrams and defini- tions concerning velocities are turned into diagrams and definitions concerning accelera- Fig. 7. tions by conceiving the diagrams as repre- senting velocities added per unit time. Thus by the addition of vector velocities and of vector accelerations, we mean the addition according to the parallelogram law. Also, according to the laws of motion a force is fully represented by the vector acceleration it produces in a body of given mass. Accordingly, forces will be said to be added when their joint effect is to be reckoned according to the parallelogram law. Hence for the fundamental vectors of DYNAMICS 57 science, namely transportations, velocities, and forces, the addition of any two of the same kind is the production of a " resultant " vector according to the rule of the parallelo- gram law. By far the simplest type of parallelogram is a rectangle, and in pure mathematics it is the relation of the single vector AC to the two component vectors, AB and AD, at right angles (cf. fig. 7), which is continually re- curring. Let x, y, and r units represent the lengths of AB, AD, and AC, and let m units of angle represent the magnitude of the angle BAG. Then the relations between #, y, r, and m, in all their many aspects are the con- tinually recurring topic of pure mathematics ; and the results are of the type required for application to the fundamental vectors of mathematical physics. This diagram is the chief bridge over which the results of pure mathematics pass in order to obtain applica- tion to the facts of nature. CHAPTER V THE SYMBOLISM OF MATHEMATICS WE now return to pure mathematics, and consider more closely the apparatus of ideas out of which the science is built. Our first concern is with the symbolism of the science, and we start with the simplest and universally known symbols, namely those of arithmetic. Let us assume for the present that we have sufficiently clear ideas about the integral numbers, represented in the Arabic notation by 0,1,2, . . ., 9, 10, 11, ... 100, 101, . . . and so on. This notation was introduced into Europe through the Arabs, but they appar- ently obtained it from Hindoo sources. The first known work * in which it is systematic- ally explained is a work by an Indian mathe- matician, Bhaskara (born 1114 A.D.). But the actual numerals can be traced back to the seventh century of our era, and perhaps were originally invented in Tibet. For our present * For the detailed historical facts relating to pure mathematics, I am chiefly indebted to A Short History of Mathematics, by W. W. B. Ball. 58 SYMBOLISM OF MATHEMATICS 59 purposes, however, the history of the notation is a detail. The interesting point to notice is the admirable illustration which this numeral system affords of the enormous im- portance of a good notation. By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the race. Before the introduction of the Arabic notation, multipli- cation was difficult, and the division even of integers called into play the highest mathe- matical faculties. Probably nothing in the modern world would have more astonished a Greek mathematician than to learn that, under the influence of compulsory education, a large proportion of the population of Western Europe could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impos- sibility. The consequential extension of the notation to decimal fractions was not accomplished till the seventeenth century. Our modern power of easy reckoning with decimal fractions is the almost miraculous result of the gradual discovery of a perfect notation. Mathematics is often considered a diffi- cult and mysterious science, because of the numerous symbols which it employs. Of course, nothing is more incomprehensible than 60 INTRODUCTION TO MATHEMATICS a symbolism which we do not understand. Also a symbolism, which we only partially understand and are unaccustomed to use, is difficult to follow. In exactly the same way the technical terms of any profession or trade are incomprehensible to those who have never been trained to use them. But this is not because they are difficult in themselves. On the contrary they have invariably been intro- duced to make things easy. So in mathe- matics, granted that we are giving any serious attention to mathematical ideas, the sym- bolism is invariably an immense simplifica- tion. It is not only of practical use, but is of great interest. For it represents an analy- sis of the ideas of the subject and an almost pictorial representation of their relations to each other. If anyone doubts the utility of symbols, let him write out in full, without any symbol whatever, the whole meaning of the following equations which represent some of the fundamental laws of algebra * : z+y=y+x (l) (x+y}+z=x+(y+z) .. . . (2) as x y=y xx (3) (x x y) x z=x x (y x z) . . (4) x x (y+z)=(x x y)+(x x z) . . (5) Here (1) and (2) are called the commutative and associative laws for addition, (3) and (4) * Cf. Note A, p. 250. SYMBOLISM OF MATHEMATICS 61 are the commutative and associative laws for multiplication, and (5) is the distributive law relating addition and multiplication. For ex- ample, without symbols, (1) becomes: If a second number be added to any given number the result is the same as if the first given number had been added to the second number. This example shows that, by the aid of sym- bolism, we can make transitions in reasoning almost mechanically by the eye, which other- wise would call into play the higher faculties of the brain. It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the num- ber of important operations which we can perform without thinking about them. Opera- tions of thought are like cavalry charges in a battle they are strictly limited in num- ber, they require fresh horses, and must only be made at decisive moments. One very important property for symbolism to possess is that it should be concise, so as to be visible at one glance of the eye and to be rapidly written. Now we cannot place sym- bols more concisely together than by placing them in immediate juxtaposition. In a good symbolism therefore, the juxtaposition of im- 02 INTRODUCTION TO MATHEMATICS portant symbols should have an important meaning. This is one of the merits of the Arabic notation for numbers ; by means of ten symbols, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and by simple juxtaposition it symbolizes any number whatever. Again in algebra, when we have two variable numbers x and y, we have to make a choice as to what shall be denoted by their juxtaposition xy. Now the two most important ideas on hand are those of addition and multiplication. Mathematicians have chosen to make their symbolism more concise by denning xy to stand for x x y. Thus the laws (3), (4), and (5) above are in general written, xy=yx, (xy}z=x(yz) y x(y+z)=xy+xz, thus securing a great gain in conciseness. The same rule of symbolism is applied to the juxtaposition of a definite number and a vari- able : we write 3x for 3 x x, and 30x for 30 x x. It is evident that in substituting definite numbers for the variables some care must be taken to restore the x, so as not to conflict with the Arabic notation. Thus when we substitute 2 for x and 3 for y in xy, we must write 2x3 for xy, and not 23 which means 20+3. It is interesting to note how important for the development of science a modest-looking symbol may be. It may stand for the em- phatic presentation of an idea, often a very SYMBOLISM OF MATHEMATICS 63 subtle idea, and by its existence make it easy to exhibit the relation of this idea to all the complex trains of ideas in which it occurs. For example, take the most modest of all symbols, namely, 0, which stands for the num- ber zero. The Roman notation for numbers had no symbol for zero, and probably most mathematicians of the ancient world would have been horribly puzzled by the idea of the number zero. For, after all, it is a very subtle idea, not at all obvious. A great deal of discussion on the meaning of the zero of quantity will be found in philosophic works. Zero is not, in real truth, more difficult or subtle in idea than the other cardinal numbers. What do we mean by 1 or by 2, or by 3 ? But we are familiar with the use of these ideas, though we should most of us be puzzled to give a clear analysis of the simpler ideas which go to form them. The point about zero is that we do not need to use it in the opera- tions of daily life. No one goes out to buy zero fish. It is in a way the most civilized of all the cardinals, and its use is only forced on us by the needs of cultivated modes of thought. Many important services are ren- dered by the symbol 0, which stands for the number zero. The symbol developed in connection with the Arabic notation for numbers of which it is an essential part. For in that notation the 64 INTRODUCTION TO MATHEMATICS value of a digit depends on the position in which it occurs. Consider, for example, the digit 5, as occurring in the numbers 25, 51, 3512, 5213. In the first number 5 stands for five, in the second number 5 stands for fifty, in the third number for five hundred, and in the fourth number for five thousand. Now, when we write the number fifty-one in the symbolic form 51, the digit 1 pushes the digit 5 along to the second place (reckoning from right to left) and thus gives it the value fifty. But when we want to symbolize fifty by itself, we can have no digit 1 to perform this service ; we want a digit in the units place to add nothing to the total and yet to push the 5 along to the second place. This service is performed by 0, the symbol for zero. It is extremely probable that the men who intro- duced for this purpose had no definite con- ception in their minds of the number zero. They simply wanted a mark to symbolize the fact that nothing was contributed by the digit's place in which it occurs. The idea of zero probably took shape gradually from a desire to assimilate the meaning of this mark to that of the marks, 1, 2, ... 9, which do re- present cardinal numbers. This would not represent the only case in which a subtle idea has been introduced into mathematics by a symbolism which in its origin was dictated by practical convenience. SYMBOLISM OF MATHEMATICS 65 Thus the first use of was to make the arable notation possible no slight service. We can imagine that when it had been intro- duced for this purpose, practical men, of the sort who dislike fanciful ideas, deprecated the silly habit of identifying it with a number zero. But they were wrong, as such men always are when they desert their proper function of masticating food which others have prepared. For the next service performed by the symbol essentially depends upon assign- ing to it the function of representing the number zero. This second symbolic use is at first sight so absurdly simple that it is difficult to make a beginner realize its importance. Let us start with a simple example. In Chapter II. we mentioned the correlation between two variable numbers x and y represented by the equation x -\-y 1. This can be represented in an indefinite number of ways ; for example, x = 1 y, y= Ix, 2x+3y 1 = x-\-2y, and so on. But the important way of stating it is x+y-I = 0. Similarly the important way of writing the equation x=I is a? 1=0, and of representing the equation 3x 2=2x* is 2x* 30+2=0. The point is that all the symbols which repre- sent variables, e.g. x and y, and the symbols 66 INTRODUCTION TO MATHEMATICS representing some definite number other than zero, such as 1 or 2 in the examples above, are written on the left-hand side, so that the whole left-hand side is equated to the number zero. The first man to do this is said to have been Thomas Harriot, born at Oxford in 1560 and died in 1621. But what is the importance of this simple symbolic pro- cedure ? It made possible the growth of the modern conception of algebraic form. This is an idea to which we shall have con- tinually to recur ; it is not going too far to say that no part of modern mathematics can be properly understood without constant re- currence to it. The conception of form is so general that it is difficult to characterize it in abstract terms. At this stage we shall do better merely to consider examples. Thus the equations 2x 3=0, #1=0, 5x 6=0, are all equations of the same form, namely, equations involving one unknown x, which is not multiplied by itself, so that a? 2 , a? 3 , etc., do not appear. Again 3a? 2 2x + 1 = 0, x 2 3x + 2 =0, o? 2 4=0, are all equations of the same form, namely, equations involving oneunknown x in which a?xa;, that is a? 2 , appears. These equations are called quadratic equations. Similarly cubic equations, in which a? 3 appears, yield another form, and so on. Among the three quadratic equations given above there is a minor difference between the last equa- SYMBOLISM OF MATHEMATICS 67 tion, x 2 4=0, and the preceding two equa- tions, due to the fact that x (as distinct from x 2 ) does not appear in the last and does in the other two. This distinction is very unimportant in comparison with the great fact that they are all three quadratic equations. Then further there are the forms of equation stating correlations between two variables; for example, x+y 1=0, 2x -\-3y- 8=0, and so on. These are examples of what is called the linear form of equation. The reason for this name of " linear " is that the graphic method of representation, which is explained at the end of Chapter II, always represents such equations by a straight line. Then there are other forms for two variables for example, the quadratic form, the cubic form, and so on. But the point which we here insist upon is that this study of form is facilitated, and, indeed, made possible, by the standard method of writing equations with the symbol on the right-hand side. There is yet another function performed by in relation to the study of form. Whatever number x may be, x x=Q, and #+0=#. By means of these properties minor differ- ences of form can be assimilated. Thus the difference mentioned above between the quad- ratic equations x 2 3#-f-2=0, and x 2 4=0, can be obliterated by writing the latter 68 INTRODUCTION TO MATHEMATICS equation in the form # 2 +(Oxo:) 4=0. For, by the laws stated above, # 2 +(Ox#) 4 = a^+O 4=# 2 4. Hence the equation # 2 4 =0, is merely representative of a particular class of quadratic equations and belongs to the same general form as does x 2 3#-f2=0. For these three reasons the symbol 0, re- presenting the number zero, is essential to modern mathematics. It has rendered pos- sible types of investigation which would have been impossible without it. The symbolism of mathematics is in truth the outcome of the general ideas which dominate the science. We have now two such general ideas before us, that of the vari- able and that of algebraic form. The junction of these concepts has imposed on mathematics another type of symbolism almost quaint in its character, but none the less effective. We have seen that an equation involving two variables, x and y, represents a particular correlation between the pair of variables. Thus x -{-y 1 =0 represents one definite corre- lation, and 3x+2y 5=0 represents another definite correlation between the variables x and y ; and both correlations have the form of what we have called linear correlations. But now, how can we represent any linear correlation between the variable numbers x and y ? Here we want to symbolize any linear correlation ; just as x symbolizes any SYMBOLISM OF MATHEMATICS 69 number. This is done by turning the numbers which occur in the definite correlation 3x+2y 5 =o into letters. We obtain ax -\-by c =0. Here a, b, c, stand for variable numbers just as do x and y : but there is a difference in the use of the two sets of variables. We study the general properties of the relationship be- tween x and y while a, b, and c have un- changed values. We do not determine what the values of a, b, and c are ; but whatever they are, they remain fixed while we study the relation between the variables x and y for the whole group of possible values of x and y. But when we have obtained the pro- perties of this correlation, we note that, be- cause a, b, and c have not in fact been deter- mined, we have proved properties which must belong to any such relation. Thus, by now varying a, b, and c, we arrive at the idea that ax+by c=0 represents a variable linear correlation between x and y. In comparison with x and t/, the three variables a, b, and c are called constants. Variables used in this way are sometimes also called parameters. Now, mathematicians habitually save the trouble of explaining which of their variables are to be treated as " constants," and which as variables, considered as correlated in their equations, by using letters at the end of the alphabet for the " variable " variables, and letters at the beginning of the alphabet for 70 INTRODUCTION TO MATHEMATICS the " constant " variables, or parameters. The two systems meet naturally about the middle of the alphabet. Sometimes a word or two of explanation is necessary ; but as a matter of fact custom and common sense are usually sufficient, and surprisingly little con- fusion is caused by a procedure which seems so lax. The result of this continual elimination of definite numbers by successive layers of para- meters is that the amount of arithmetic per- formed by mathematicians is extremely small. Many mathematicians dislike all numerical computation and are not particularly expert at it. The territory of arithmetic ends where the two ideas of "variables" and of "alge- braic form " commence their sway. CHAPTER VI GENERALIZATIONS OT NUMBER ONE great peculiarity of mathematics is the set of allied ideas which have been invented in connection with the integral numbers from which we started. These ideas may be called extensions or generalizations of number. In the first place there is the idea of fractions. The earliest treatise on arithmetic which we possess was written by an Egyptian priest, named Ahmes, between 1700 B.C. and 1100 B.C., and it is probably a copy of a much older work. It deals largely with the properties of fractions. It appears, therefore, that this concept was developed very early in the his- tory of mathematics. Indeed the subject is a very obvious one. To divide a field into three equal parts, and to take two of the parts, must be a type of operation which had often occurred. Accordingly, we need not be surprised that the men of remote civilizations were familiar with the idea of two-thirds, and 71 72 INTRODUCTION TO MATHEMATICS with allied notions. Thus as the first genera- lization of number we place the concept of fractions. The Greeks thought of this sub- ject rather in the form of ratio, so that a Greek would naturally say that a line of two feet in length bears to a line of three feet in length the ratio of 2 to 8. Under the influence of our algebraic notation we would more often say that one line was two-thirds of the other in length, and would think of two-thirds as a numerical mul- tiplier. In connection with the theory of ratio, or fractions, the Greeks made a great discovery, which has been the occasion of a large amount of philosophical as well as mathematical thought. They found out the existence of " incommensurable " ratios. They proved, in fact, during the course of their geometrical investigations that, starting with a line of any length, other lines must exist whose lengths do not bear to the original length the ratio of any pair of integers or, in other words, that lengths exist which are not any exact fraction of the original length. For example, the diagonal of a square cannot be expressed as any fraction of the side of the same square ; in our modern notation the length of the diagonal is \/2 times the length of the side. But there is no fraction which exactly represents \/2. We can approximate GENERALIZATIONS OF NUMBERS 73 to V2 as closely as we like, but we never exactly reach its value. For example, is ' 25 g just less than 2, and - is greater than 2, so 4 7 3 that \/2 lies between -= and 5 . But the best o 1. systematic way of approximating to \/2 in obtaining a series of decimal fractions, each bigger than the last, is by the ordinary method of extracting the square root ; thus the series 14 141 1414 18 * 16' iob' looo' and so on ' Ratios of this sort are called by the Greeks incommensurable. They have excited from the time of the Greeks onwards a great deal of philosophic discussion, and the difficulties connected with them have only recently been cleared up. We will put the incommensurable ratios with the fractions, and consider the whole set of integral numbers, fractional numbers, and incommensurable numbers as forming one class of numbers which we will call " real numbers." We always think of the real numbers as arranged in order of magnitude, starting from zero and going upwards, and becoming indefinitely larger and larger as we proceed. The real numbers are conveniently represented by points on a line. Let OX be I 2 1 3 2 2 5 2 3 7 2 4 o M A N B P C Q D X any line bounded at O and stretching away in- definitely in the direction OX. Take any con- venieiit point, A, on it, so that OA represents the unit length ; and divide off lengths AB, BC, CD, and so on, each equal to OA. Then the point O represents the number 0, A the number 1, B the number 2, and so on. In fact the number represented by any point is the measure of its distance from O, in terms of the unit length OA. The points between O and A represent the proper fractions and the incommensurable numbers less than 1 ; the middle point of OA represents -, that of 2 o e AB represents -, that of BC represents -, and a a so on. In this way every point on OX repre- sents some one real number, and every real number is represented by some one point on OX. The series (or row) of points along OX, starting from O and moving regularly in the direction from to X, represents the real numbers as arranged in an ascending order GENERALIZATIONS OF NUMBERS 75 of size, starting from zero and continually increasing as we go on. All this seems simple enough, but even at this stage there are some interesting ideas to be got at by dwelling on these obvious facts. Consider the series of points which represent the integral numbers only, namely, the points, O, A, B, C, D, etc. Here there is a first point 0, a definite next point, A, and each point, such as A or B, has one definite immediate predecessor and one definite immediate suc- cessor, with the exception of 0, which has no predecessor ; also the series goes on in- definitely without end. This sort of order is called the type of order of the integers ; its essence is the possession of next-door neigh- bours on either side with the exception of No. 1 in the row. Again consider the integers and fractions together, omitting the points which correspond to the incommensurable ratios. The sort of serial order which we now obtain is quite different. There is a first term ; but no term has any immediate pre- decessor or immediate successor. This is easily seen to be the case, for between any two fractions we can always find another fraction intermediate in value. One very simple way of doing this is to add the fractions together and to halve the result. For ex- ample, between f and , the fraction (f + f), that is IT, lies ; and between f and H the 76 INTRODUCTION TO MATHEMATICS fraction (f -j- {), that is If, lies ; and so on indefinitely. Because of this property the series is said to be " compact." There is no end point to the series, which increases in- definitely without limit as we go along the line OX. It would seem at first sight as though the type of series got in this way from the fractions, always including the integers, would be the same as that got from all the real numbers, integers, fractions, and incom- mensurables taken together, that is, from all the points on the line OX. All that we have hitherto said about the series of fractions applies equally well to the series of all real numbers. But there are important differ- ences which we now proceed to develop. The absence of the incommensurables from the series of fractions leaves an absence of end- points to certain classes. Thus, consider the incommensurable A/2. In the series of real numbers this stands between all the numbers whose squares are less than 2, and all the numbers whose squares are greater than 2. But keeping to the series of fractions alone and not thinking of the incommensurables, so that we cannot bring in \/2, there is no frac- tion which has the property of dividing off the series into two parts in this way, i.e. so that all the members on one side have their squares less than 2, and on the other side greater than 2. Hence in the series of frac- GENERALIZATIONS OF NUMBERS 77 tions there is a quasi-gap where \/2 ought to come. This presence of quasi-gaps in the series of fractions may seem a small matter ; but any mathematician, who happens to read this, knows that the possible absence of limits or maxima to a class of numbers, which yet does not spread over the whole series of num- bers, is no small evil. It is to avoid this difficulty that recourse is had to the incom- mensurables, so as to obtain a complete series with no gaps. There is another even more fundamental difference between the two series. We can rearrange the fractions in a series like that of the integers, that is, with a first term, and such that each term has an immediate suc- cessor and (except the first term) an immediate predecessor. We can show how this can be done. Let every term in the series of fractions and integers be written in the fractional form by writing j for 1, f for 2, and so on for all the integers, excluding 0. Also for the moment we will reckon fractions which are equal in value but not reduced to their lowest terms as distinct ; so that, for example, until further notice f, |, J-, A> etc., are all reckoned as dis- tinct. Now group the fractions into classes by adding together the numerator and de- nominator of each term. For the sake of brevity call this sum of the numerator and denominator of a fraction its index. Thus 7 78 INTRODUCTION TO MATHEMATICS is the index of |, and also of f , and of f . Let the fractions in each class be all fractions which have some specified index, which may therefore also be called the class index. Now arrange these classes in the order of magni- tude of their indices. The first class has the index 2, and its only member is T 5 the second class has the index 3, and its members are and f ; the third class has the index 4, and its members are J, f, f; the fourth class has the index 5, and its members are i i> f T J an d so on - It is easy to see that the number of members (still including frac- tions not in their lowest terms) belonging to any class is one less than its index. Also the members of any one class can be arranged in order by taking the first member to be the fraction with numerator 1, the second mem- ber to have the numerator 2, and so on, up to (n 1) where n is the index. Thus for the class of index n, the members appear in the order. 123 n-1 . ~, -, -, . . ., = . The mem- n 1 n 2 n3 1 bers of the first four classes have in fact been mentioned in this order. Thus the whole set of fractions have now been arranged in an order like that of the integers. It runs thus 1121 T21 31234 r 2' r ' LLl' r ? 3* 2' r GENERALIZATIONS OF NUMBERS 79 n-2 _1 2_ 3 n-1 1 1 'ra-l'n-2'n-3' ' '' 1 ' n' and so on. Now we can get rid of all repetitions of fractions of the same value by simply striking them out whenever they appear after their first occurrence. In the few initial terms written down above, f which is enclosed above in square brackets is the only fraction not in its lowest terms. It has occurred before as T. Thus this must be struck out. But the series is still left with the same properties, namely, (a) there is a first term, (6) each term has next-door neighbours, (c) the series goes on without end. It can be proved that it is not possible to arrange the whole series of real numbers in this way. This curious fact was discovered by Georg Cantor, a German mathematician still living ; it is of the utmost importance in the philosophy of mathematical ideas. We are here in fact touching on the fringe of the great problems of the meaning of continuity and of infinity. Another extension of number comes from the introduction of the idea of what has been variously named an operation or a step, names which are respectively appropriate from slightly different points of view. We will start with a particular case. Consider 80 INTRODUCTION TO MATHEMATICS the statement 2+3=5. We add 3 to 2 and obtain 5. Think of the operation of adding 3: let this be denoted by +3. Again 43 =1. Think of the operation of subtracting 3 : let this be denoted by 3. Thus instead of considering the real numbers in themselves, we consider the operations of adding or sub- tracting them : instead of -v/2, we consider + V2 and \/2, namely the operations of adding V2 and of subtracting \/2. Then we can add these operations, of course in a different sense of addition to that in which we add numbers. The sum of two operations is the single operation which has the same effect as the two operations applied successively. In what order are the two operations to be applied ? The answer is that it is indifferent, since for example 2+3+1=2+1+3; so that the addition of the steps +3 and +1 is commutative. Mathematicians have a habit, which is puzzling to those engaged in tracing out meanings, but is very convenient in practice, of using the same symbol in different though allied senses. The one essential requisite for a symbol in their eyes is that, whatever its possible varieties of meaning, the formal laws for its use shall always be the same. In GENERALIZATIONS OF NUMBERS 81 accordance with this habit the addition of operations is denoted by -f as well as the addition of numbers. Accordingly we can write where the middle + on the left-hand side denotes the addition of the operations -{-3 and -f 1. But, furthermore, we need not be so very pedantic in our symbolism, except in the rare instances when we are directly tracing meanings ; thus we always drop the first + of a line and the brackets, and never write two + signs running. So the above equation becomes 3+1=4, which we interpret as simple numerical addi- tion, or as the more elaborate addition of operations which is fully expressed in the previous way of writing the equation, or lastly as expressing the result of applying the operation +1 to the number 3 and ob- taining the number 4. Any interpretation which is possible is always correct. But the only interpretation which is always possible, under certain conditions, is that of operations. The other interpretations often give non- sensical results. This leads us at once to a question, which must have been rising insistently in the reader's mind : What is the use of all this elaboration ? At this point our friend, the practical man, will surely step in and insist on sweeping away all these silly cobwebs of the brain. The answer is that what the mathe- matician is seeking is Generality. This is an idea worthy to be placed beside the notions of the Variable and of Form so far as concerns its importance in governing mathematical procedure. Any limitation whatsoever upon the generality of theorems, or of proofs, or of interpretation is abhorrent to the mathe- matical instinct. These three notions, of the variable, of form, and of generality, compose a sort of mathematical trinity which preside over the whole subject. They all really spring from the same root, namely from the abstract nature of the science. Let us see how generality is gained by the introduction of this idea of operations. Take the equation x +1=3; the solution is x =2. Here we can interpret our symbols as mere numbers, and the recourse to " operations " is entirely unnecessary. But, if a? is a mere number, the equation #+3=1 is nonsense. For x should be the number of things which remain when you have taken 3 things away from 1 thing ; and no such procedure is possible. At this point our idea of algebraic form steps in, itself only generalization under another aspect. We consider, therefore, the GENERALIZATIONS OF NUMBERS 83 general equation of the same form as <r+l =3, This equation is x-\-a=b y and its solution is x =b a. Here our difficulties become acute ; for this form can only be used for the numeri- cal interpretation so long as b is greater than a, and we cannot say without qualification that a and b may be any constants. In other words we have introduced a limitation on the variability of the " constants " a and b, which we must drag like a chain throughout all our reasoning. Really prolonged mathe- matical investigations would be impossible under such conditions. Every equation would at last be buried under a pile of limita- tions. But if we now interpret our symbols as " operations," all limitation vanishes like magic. The equation x +1=3 gives # = +2, the equation x +3=1 gives x= 2, the equa- tion x-\~a=b gives x=ba which is an opera- tion of addition or subtraction as the case may be. We need never decide whether b a represents the operation of addition or of subtraction, for the rules of procedure with the symbols are the same in either case. It does not fall within the plan of this work 1 to write a detailed chapter of elementary algebra. Our object is merely to make plain the fundamental ideas which guide the forma- tion of the science. Accordingly we do not further explain the detailed rules by which the " positive and negative numbers " are 84 INTRODUCTION TO MATHEMATICS multiplied and otherwise combined. We have explained above that positive and negative numbers are operations. They have also been called " steps." Thus +3 is the step by which we go from 2 to 5, and 3 is the step backwards by which we go from 5 to 2. Consider the line OX divided in the way ex- plained in the earlier part of the chapter, so that its points represent numbers. Then +2 , D' C' B' A' +1 +2 +3 -3-2-1 A B C D E is the step from to B, or from A to C, or (if the divisions are taken backwards along OX') from C' to A', or from D f to B', and so on. Similarly 2 is the step from to B', or from B' to D', or from B to 0, or from C to A. We may consider the point which is reached by a step from 0, as representative of that step. Thus A represents +1> B represents +2, A' represents 1, B' represents 2, and so on. It will be noted that, whereas previ- ously with the mere "unsigned " real numbers the points on one side of only, namely along OX, were representative of numbers, now with steps every point on the whole line stretching on both sides of is representative of a step. This is a pictorial representation of the superior generality introduced by the positive and negative numbers, namely the GENERALIZATIONS OF NUMBERS 85 operations or steps. These "signed " num- bers are also particular cases of what have been called vectors (from the Latin veho, I draw or carry). For we may think of a particle as carried from O to A, or from A to B. In suggesting a few pages ago that the practical man would object to the subtlety involved by the introduction of the positive and negative numbers, we were libelling that excellent individual. For in truth we are on the scene of one of his greatest triumphs. If the truth must be confessed, it was the practi- cal man himself who first employed the actual symbols -f- and . Their origin is not very certain, but it seems most probable that they arose from the marks chalked on chests of goods in German warehouses, to denote excess or defect from some standard weight. The earliest notice of them occurs in a book pub- lished at Leipzig, in A.D. 1489. They seem first to have been employed in mathematics by a German mathematician, Stifel, in a book published at Nuremburg in 1544 A.D. But then it is only recently that the Germans have come to be looked on as emphatically a practical nation. There is an old epigram which assigns the empire of the sea to the English, of the land to the French, and of the clouds to the Germans. Surely it was from the clouds that the Germans fetched + and 86 INTRODUCTION TO MATHEMATICS ; the ideas which these symbols have generated are much too important for the welfare of humanity to have come from the sea or from the land. The possibilities of application of the posi- tive and negative numbers are very obvious. If lengths in one direction are represented by positive numbers, those in the opposite direction are represented by negative numbers. If a velocity in one direction is positive, that in the opposite direction is negative. If a rotation round a dial in the opposite direction to the hands of a clock (anti-clockwise) is positive, that in the clockwise direction is negative. If a balance at the bank is posi- tive, an overdraft is negative. If vitreous electrification is positive, resinous electrifica- tion is negative. Indeed, in this latter case, the terms positive electrification and negative electrification, considered as mere names, have practically driven out the other terms. An endless series of examples could be given. The idea of positive and negative numbers has been practically the most successful of mathematical subtleties. CHAPTER VII IMAGINARY NUMBERS IF the mathematical ideas dealt with in the last chapter have been a popular success, those of the present chapter have excited almost as much general attention. But their success has been of a different character, it has been what the French term a succes de scandale. Not only the practical man, but also men of letters and philosophers have ex- pressed their bewilderment at the devotion of mathematicians to mysterious entities which by their very name are confessed to be imaginary. At this point it may be useful to observe that a certain type of intellect is always worrying itself and others by discussion as to the applicability of technical terms. Are the incommensurable numbers properly called numbers ? Are the positive and negative numbers really numbers ? Are the imaginary numbers imaginary, and are they numbers ? are types of such futile questions. Now, it cannot be too clearly understood that, in science, technical terms are names arbitrarily assigned, like Christian 87 88 INTRODUCTION TO MATHEMATICS names to children. There can be no question of the names being right or wrong. They may be judicious or injudicious ; for they can sometimes be so arranged as to be easy to remember, or so as to suggest relevant and important ideas. But the essential principle involved was quite clearly enunciated in Wonderland to Alice by Humpty Dumpty, when he told her, a propos of his use of words, " I pay them extra and make them mean what I like." So we will not bother as to whether imaginary numbers are imaginary, or as to whether they are numbers, but will take the phrase as the arbitrary name of a certain mathematical idea, which we will now endeavour to make plain. The origin of the conception is in every way similar to that of the positive and nega- tive numbers. In exactly the same way it is due to the three great mathematical ideas of the variable, of algebraic form, and of generalization. The positive and negative numbers arose from the consideration of equations like #-fl=3, +3=l, and the general form x+a=b. Similarly the origin of imaginary numbers is due to equations like # 2 +l=8, a? 2 +3=1, and OJ 2 -fa=&. Exactly the same process is gone through. The equa- tion a; 2 +1 =3 becomes # 2 =2, and this has two solutions, either x = + V%, or x = V2. The statement that there are these alternative IMAGINARY NUMBERS 89 solutions is usually written x = A/2. So far all is plain sailing, as it was in the previous case. But now an analogous difficulty arises. For the equation x 2 +3=1 gives # 2 = 2 and there is no positive or negative number which, when multiplied by itself, will give a negative square. Hence, if our symbols are to mean the ordinary positive or negative numbers, there is no solution to x 2 = 2, and the equa- tion is in fact nonsense. Thus, finally taking the general form x 2 +a=b ) we find the pair of solutions x = \/(b a), when, and only when, b is not less than a. Accordingly we cannot say unrestrictedly that the " con- stants " a and b may be any numbers, that is, the " constants " a and b are not, as they ought to be, independent unrestricted " vari- ables " ; and so again a host of limitations and restrictions will accumulate round our work as we proc'eed. The same task as before therefore awaits us : we must give a new interpretation to our symbols, so that the solutions V(6 a) for the equation x 2 -{-a=b always have meaning. In other words, we require an interpretation of the symbols so that Va always has meaning whether a be positive or negative. Of course, the interpretation must be such that all the ordinary formal laws for addition, sub- traction, multiplication, and division hold good ; and also it must not interfere with the 90 INTRODUCTION TO MATHEMATICS generality which we have attained by the use of the positive and negative numbers. In fact, it must in a sense include them as special cases. When a is negative we may write c 2 for it, so that c 2 is positive. Then Va = V(^?) =y r ((->l)xc 2 } =V(-i) Vc*=c vT^i)- Hence, if we can so interpret our symbols that V( 1) has a meaning, we have attained our object. Thus V( 1) has come to be looked on as the head and forefront of all the imaginary quantities. This business of finding an interpretation for V( 1) is a much tougher job than the analogous one of interpreting 1. In fact, while the easier problem was solved almost instinctively as soon as it arose, it at first hardly occurred, even to the greatest mathe- maticians, that here a problem existed which was perhaps capable of solution. Equations like x 2 - 3, when they arose, were simply ruled aside as nonsense. However, it came to be gradually perceived during the eighteenth century, and even earlier, how very convenient it would be if an interpretation could be assigned to these nonsensical symbols. Formal reasoning with these symbols was gone through, merely assuming that they obeyed the ordinary IMAGINARY NUMBERS 91 algebraic laws of transformation ; and it was seen that a whole world of interesting results could be attained, if only these symbols might legitimately be used. Many mathematicians were not then very clear as to the logic of their procedure, and an idea gained ground that, in some mysterious way, symbols which mean nothing can by appropriate manipula- tion yield valid proofs of propositions. No- thing can be more mistaken. A symbol which has not been properly defined is not a symbol at all. It is merely a blot of ink on paper which has an easily recognized shape. Nothing can be proved by a succession of blots, except the existence of a bad pen or a careless writer. It was during this epoch that the epithet "imaginary" came to be applied to V( 1). What these mathema- ticians had really succeeded in proving were a series of hypothetical propositions, of which this is the blank form: If interpretations exist for V( 1) and for the addition, sub- traction, multiplication, and division of V( 1) which make the ordinary algebraic rules (e.g. x+y=y-\-x, etc.) to be satisfied, then such and such results follows. It was natural that the mathematicians should not always appreciate the big "If," which ought to have preceded the statements of their re- suits. As may be expected the interpretation, 92 INTRODUCTION TO MATHEMATICS when found, was a much more elaborate affair than that of the negative numbers and the reader's attention must be asked for some careful preliminary explanation. We have already come across the representation of a point by two numbers. By the aid of the p' M 1 p" y' A'' y' ptu y Fig. 8. positive and negative numbers we can now represent the position of any point in a plane by a pair of such numbers. Thus we take the pair of straight lines XOX' and YOY' t at right angles, as the " axes " from which we start all our measurements. Lengths mea- sured along OX and OY are positive, and measured backwards along OX' and OY' are negative. Suppose that a pair of numbers, written in order,e.g. (+3, +!) so that there IMAGINARY NUMBERS 93 is a first number (+3 in the above example), and a second number (+1 in the above ex- ample), represents measurements from O along XOX' for the first number, and along YOY' for the second number. Thus (cf . fig. 9) in ( +3, +1) a length of 3 units is to be measured along XOX' in the positive direction, that is from O towards X, and a length +1 measured along YOY' in the positive direc~ tion, that is from O towards F. Similarly in (3, +1) the length of 3 units is to be measured from O towards X', and of 1 unit from towards Y . Also in (3, 1) the two lengths are to be measured along OX' and OF' respectively, and in (+3, 1) along OX and OF* respectively. Let us for the moment call such a pair of numbers an " ordered couple." Then, from the two num- bers 1 and 3, eight ordered couples can be generated, namely (+1, +3), (-1, +3), (-1, -3), (+1, -3), (+3, +1), (-3, +1), (-3, -1), (+3, -1). Each of these eight "ordered couples " directs a process of measurement along XOX' and FOF' which is different from that directed by any of the others. The processes of measurement represented by the last four ordered couples, mentioned above, are given pictorially in the figure. The lengths OM and ON together correspond 94 INTRODUCTION TO MATHEMATICS to (+3, +1), the lengths OM' and ON together correspond to (3, +1), OM' and ON' together to (3, 1), and OM and ON' together to (+3, 1). But by com- pleting the various rectangles, it is easy to see that the point P completely determines and is determined by the ordered couple P' N P fl . -3 *3 M H' -' P" N' y.' Fig. 9. (+3, +1), the point P' by (-3, -fl), the point P" by (-3, -1), and the point P'" by (+3, 1). More generally in the previous figure (8), the point P corresponds to the ordered couple (x, y), where x and y in the figure are both assumed to be positive, the point P' corresponds to (#', y), where x' in the figure is assumed to be negative, P" to (#' 2/')> and P'" to (x, y'). Thus an ordered IMAGINARY NUMBERS 95 couple (x, y), where x and y are any positive or negative numbers, and the corresponding point reciprocally determine each other. It is convenient to introduce some names at this juncture. In the ordered couple (x, y) the first number x is called the " abscissa " of the corresponding point, and the second number y is called the " ordinate " of the point, and the two numbers together are called the " co- ordinates " of the point. The idea of deter- mining the position of a point by its " co- ordinates " was by no means new when the theory of " imaginaries " was being formed. It was due to Descartes, the great French mathematician and philosopher, and appears in his Discours published at Leyden in 1637 A.D. The idea of the ordered couple as a thing on its own account is of later growth and is the outcome of the efforts to interpret imaginaries in the most abstract way possible. It may be noticed as a further illustration of this idea of the ordered couple, that the point M in fig. 9 is the couple (+3, 0), the point N is the couple (0, +1), the point M' the couple (3, 0), the point N' the couple (0, 1), the point O the couple (0, 0). Another way of representing the ordered couple (x, y) is to think of it as representing the dotted line OP (cf . fig. 8), rather than the point P. Thus the ordered couple represents a line drawn from an " origin," O, of a certain 96 INTRODUCTION TO MATHEMATICS length and in a certain direction. The line OP may be called the vector line from O to P, or the step from O to P. We see, therefore, that we have in this chapter only extended the interpretation which we gave formerly of the positive and negative numbers. This method of representation by vectors is very useful when we consider the meaning to be assigned to the operations of the addition and multiplication of ordered couples. We will now go on to this question, and ask what meaning we shall find it convenient to assign to the addition of the two ordered couples (x, y) and (a?', y'). The interpreta- tion must, (a) make the result of addition to be another ordered couple, (b) make the operation commutative so that (x, y) + (x', y') =(%'Ty') -f-(tf71/)> (c) make the opera- tion associative so that (d) make the result of subtraction unique, so that when we seek to determine the unknown ordered couple (x, y) so as to satisfy the equation (x, y)+(a, b)=(c, d), there is one and only one answer which we can represent by (x, y)=(c, d)-(a t b). IMAGINARY NUMBERS 97 All these requisites are satisfied by taking (x y y)+(x', y') to mean the ordered couple (x+x', y+y'). Accordingly by definition we put (*, t/)+(a?', y')=(x+x f , y+y'). Notice that here we have adopted the mathe- matical habit of using the same symbol -f- in different senses. The + on the left-hand side of the equation has the new meaning of + which we are just defining ; while the two -f-'s on the right-hand side have the meaning of the addition of positive and negative num- bers (operations) which was defined in the last chapter. No practical confusion arises from this double use. As examples of addition we have (+3, + l)+(+2, + 6) =(+5, + 7), (+3, - l)+(-2, - 6)=(+l - 7), ( +8,+l)+(-8, -1)=(0, 0). The meaning of subtraction is now settled for us. We find that (x, y)(u, V)=(IK-U, yv). Thus { +3, + 2) -( +1, + 1) =( +2, -f 1), and ( +1, - 2) -( +2, - 4)=( -1, + 2), and ( -1, - 2) -( +2, + 3) =( -3, - 5). D 98 INTRODUCTION TO MATHEMATICS It is easy to see that (x, y)-(u, v)=(x, */)+(-, -v). Also 0* y) (# y) = (> o). Hence (0, 0) is to be looked on as the zero ordered couple. For example ( S/)-H 0)=(a?, y). The pictorial representation of the addition of ordered couples is surprisingly easy. y ....-.i K 4 1 r .-*7 "' F ^ / S M, M H' Fig. 10. Let OP represent (x, y) so that OM=x and PM=y ; let OQ represent (x\, y\) so that OMi=Xi&nd QMi=t/i. Complete the paral- lelogram OPRQ by the dotted lines PR and QR, then the diagonal OR is the ordered couple (x-\-Xi, j/+t/i). For draw PS parallel IMAGINARY NUMBERS 99 to OX ; then evidently the triangles OQM i and PRS are in all respects equal. Hence MM'=PS=Xi, and RS=QM l \ and there- fore OM'=OM+MM' = RM' =SM' +RS = Thus OJ? represents the ordered couple as required. This figure can also be drawn with OP and OQ in other quadrants. It is at once obvious that we have here come back to the parallelogram law, which was mentioned in Chapter VI. , on the laws of motion, as applying to velocities and forces. It will be remembered that, if OP and OQ represent two velocities, a particle is said to be moving with a velocity equal to the two velocities added together if it be moving with the velocity OR. In other words OR is said to be the resultant of the two velocities OP and OQ. Again forces acting at a point of a body can be represented by lines just as velocities can be ; and the same parallelogram law holds, namely, that the resultant of the two forces OP and OQ is the force represented by the diagonal OR. It follows that we can look on an ordered couple as representing a velocity or a force, and the rule which we have just given for the addition of ordered couples then represents the fundamental laws of mechanics for the addition of forces and 100 INTRODUCTION TO MATHEMATICS velocities. One of the most fascinating characteristics of mathematics is the surpris- ing way in which the ideas and results of different parts of the subject dovetail into each other. During the discussions of this and the previous chapter we have been guided merely by the most abstract of pure mathe- matical considerations ; and yet at the end of them we have been led back to the most fundamental of all the laws of nature, laws which have to be in the mind of every engineer as he designs an engine, and of every naval architect as he calculates the stability of a ship. It is no paradox to say that in our most theoretical moods we may be nearest to our most practical applications* CHAPTER VIII IMAGINARY NUMBERS (Continued) THE definition of the multiplication of ordered couples is guided by exactly the same considerations as is that of their addition. The interpretation of multiplication must be such that (a) the result is another ordered couple, (/3) the operation is commutative, so that (x, y) x(o?', y')=(x', y'} x(#, t/), (7) the operation is associative, so that {(x, y)x(x f , y')} x (u, v) =(# J/)x{(' y')*(u, v)} 9 (&) must make the result of division unique [with an exception for the case of the zero couple (0, 0)], so that when we seek to deter- mine the unknown couple (x, y) so as to satisfy the equation (x, y)x(a, b)=(c, d), there is one and only one answer, which we can represent by (*, S/)= (c, d)-H(a, 6), or by (, t/)= JA*> 101 102 INTRODUCTION TO MATHEMATICS (e) Furthermore the law involving both addition and multiplication, called the dis- tributive law, must be satisfied, namely (x,y)x{(a,b)+(c 9 d)} = {(*, y) x (a, 6)} +{(*, y) x (c, d)}. All these conditions (a), (), (7), (8), (e) can be satisfied by an interpretation which, though it looks complicated at first, is capable of a simple geometrical interpretation. By definition we put (x, y)x(x', y') = {(xx'-yy')> (xy' + x'y}} (A) This is the definition of the meaning of the symbol x when it is written between two ordered couples. It follows evidently from this definition that the result of multiplica- tion is another ordered couple, and that the value of the right-hand side of equation (A) is not altered by simultaneously interchanging x with x' t and y with y'. Hence conditions (a) and (/?) are evidently satisfied. The proof of the satisfaction of (7), (8), (e) is equally easy when we have given the geometrical interpretation, which we will proceed to do in a moment. But before doing this it will be interesting to pause and see whether we have attained the object for which all this elaboration was initiated. We came across equations of the form a? 2 = 3, to which no solutions could be IMAGINARY NUMBERS 103 assigned in terms of positive and negative real numbers. We then found that all our diffi- culties would vanish if we could interpret the equation x 2 = 1, i.e., if we could so define V(-l) that V( 1) x V( 1)= 1. Now let us consider the three special ordered couples * (0,0), (1,0), and (0,1). We have already proved that (*, */)+(<>, 0)=(a, y). Furthermore we now have (x, t/)x(0, 0)=(0, 0). Hence both for addition and for multiplica- tion the couple (0,0) plays the part of zero in elementary arithmetic and algebra ; com- pare the above equations with c+0=a?, and x x 0=0. Again consider (1, 0) : this plays the part of 1 in elementary arithmetic and algebra. In these elementary sciences the special characteristic of 1 is that x xl=#, for all values of x. Now by our law of multiplica- tion (x t y) x (1, 0)= {(x- 0), (y +0)} = (*, y). Thus (1, 0) is the unit couple. * For the future we follow the custom of omitting the -f sign wherever possible, thus (1,0) stands for ( + 1,0) and (0,1) for (0, 104 INTRODUCTION TO MATHEMATICS Finally consider (0,1) : this will interpret for us the symbol V( 1). The symbol must therefore possess the characteristic property that V( 1) x V( Z T)= 1. Now by the law of multiplication for ordered couples (0,1) x (0,1) = {(0 - 1), (0 + 0)} = ( -1, 0). But (1,0) is the unit couple, and (1, 0) is the negative unit couple ; so that (0,1) has the desired property. There are, however, two roots of 1 to be provided for, namely V( 1). Consider (0, 1) ; here again re- membering that (1) 2 =1, we find, (0, 1) x(0,-l) = (-l, 0). Thus (0, 1) is the other square root of V( 1). Accordingly the ordered couples (0,1) and (0, 1) are the interpretations of V( 1) in terms of ordered couples. But which corresponds to which ? Does (0,1) correspond to + V( 1) and (0, 1) to - VC^T), or (0,1) to -Vr^andfO, - 1) to + V( 1) ? The answer is that it is per- fectly indifferent which symbolism we adopt. The ordered couples can be divided into three types, (i) the " complex imaginary " type (x,y), in which neither x nor y is zero ; (ii) the " real " type (#,0) ; (iii) the " pure imaginary " type (O,?/). Let us consider the relations of these types to each other. First multiply together the " complex imaginary ' IMAGINARY NUMBERS 105 eouple (x,y) and the " real " couple (a,0), we find (a,0)x(x,y)=(ax, ay). Thus the effect is merely to multiply each term of the couple (x,y) by the positive or negative real number a. Secondly, multiply together the " complex imaginary " couple (x,y) and the " pure imaginary " couple (0,6), we find (0,6) x(x,y)=(by, bx). Here the effect is more complicated, and is best comprehended in the geometrical inter- pretation to which we proceed after noting three yet more special cases. Thirdly, we multiply the " real " couple (o,0) by the imaginary (0,6) and obtain (a,0) x (0,6) =(0,ai). Fourthly, we multiply the two " real " couples (a,0) and (a', 0) and obtain (a,0)x(a',0)=(oa',0). Fifthly, we multiply the two "imaginary couples " (0,6) and (0, 6) and obtain (0,6)x(0,6')=(-66', 0). We now turn to the geometrical interpreta- tion, beginning first with some special cases. 106 INTRODUCTION TO MATHEMATICS Take the couples (1,3) and (2,0) and consider the equation (2,0) x (1,3) =(2,6) n u>. / if R N,/ ^^^ X 1 _^\ t X M, M N Fig. 11. In the diagram (fig. 11) the vector OP re- presents (1, 3), and the vector ON represents (2,0), and the vector OQ represents (2,6). Thus the product (2,0) x(l,3) is found geo- metrically by taking the length of the vector OQ to be the product of the lengths of the vectors OP and ON, and (in this case) by producing OP to Q to be of the required length. Again, consider the product (0,2) x (1,3), we have (0, 2)x(l, 3)=(-6, 2) The vector ONi, corresponds to (0, 2) and the vector OR to (-6,2). Thus OR which IMAGINARY NUMBERS 107 represents the new product is at right angles to OQ and of the same length. Notice that we have the same law regulating the length of OQ as in the previous case, namely, that its length is the product of the lengths of the two vectors which are multiplied to- gether ; but now that we have ONi along the *' ordinate " axis OY, instead of ON along the ** abscissa " axis OX, the direction of OP has been turned through a right- angle. Hitherto in these examples of multiplication we have looked on the vector OP as modified by the vectors ON and ONi. We shall get a clue to the general law for the direction by inverting the way of thought, and by think- ing of the vectors ON and ONi as modified by the vector OP. The law for the length re- mains unaffected ; the resultant length is the length of the product of the two vectors. The new direction for the enlarged ON (i.e. OQ) is found by rotating it in the (anti-clock- wise) direction of rotation from OX towards OY through an angle equal to the angle XOP : it is an accident of this particular case that this rotation makes OQ lie along the line OP. Again consider the product of ONi and OP ; the new direction for the enlarged ONi (i.e. OR) is found by rotating ON in the anti- clockwise direction of rotation through an angle equal to the angle XOP, namely, the angle NiOR is equal to the angle XOP. 108 INTRODUCTION TO MATHEMATICS The general rule for the geometrical repre- sentation of multiplication can now be enunci- ated thus : Fig. 12. The product of the two vectors OP and OQ is a vector OR, whose length is the pro- duct of the lengths of OP and OQ and whose direction OR is such that the angle XOR is equal to the sum of the angles XOP and XOQ. Hence we can conceive the vector OP as making the vector OQ rotate through an angle XOP (i.e. the angle QO.K = the angle XOP), or the vector OQ as making the vector OP rotate through the angle XOQ (i.e. the angle POR =the angle XOQ). We do not prove this general law, as we IMAGINARY NUMBERS 109 should thereby be led into more technical processes of mathematics than falls within the design of this book. But now we can im- mediately see that the associative law [num- bered (7) above] for multiplication is satisfied. Consider first the length of the resultant vector ; this is got by the ordinary process of multiplication for real numbers ; and thus the associative law holds for it. Again, the direction of the resultant vector is got by the mere addition of angles, and the associative law holds for this process also. So much for multiplication. We have now rapidly indicated, by considering addition and multiplication, how an algebra or " calculus " of vectors in one plane can be constructed, which is such that any two vectors in the plane can be added, or subtracted, and can be multiplied, or divided one by the other. We have not considered the technical de- tails of all these processes because it would lead us too far into mathematical details ; but we have shown the general mode of pro- cedure. When we are interpreting our alge- braic symbols in this way, we are said to be employing " imaginary quantities " or *' com- plex quantities." These terms are mere details, and we have far too much to think about to stop to enquire whether they are or are not very happily chosen. The nett result of our investigations is that 110 INTRODUCTION TO MATHEMATICS any equations like #+3=2 or (#+3) 2 = 2 can now always be interpreted into terms of vectors, and solutions found for them. In seeking for such interpretations it is well to note that 3 becomes (3,0), and 2 becomes (2,0), and x becomes the "unknown" couple (u, v) : so the two equations become respectively (u, v) +(3,0) =(2,0), and {(u,v) +(3,0)}2 = (-2,0). We have now completely solved the initial difficulties which caught our eye as soon as we considered even the elements of algebra. The science as it emerges from the solution is much more complex in ideas than that with which we started. We have, in fact, created a new and entirely different science, which will serve all the purposes for which the old science was invented and many more in addi- tion. But, before we can congratulate our- selves on this result to our labours, we must allay a suspicion which ought by this tune to have arisen in the mind of the student. The question which the reader ought to be asking himself is : Where is all this invention of new interpretations going to end ? It is true that we have succeeded in interpreting algebra so as always to be able to solve a quadratic equation like a? 2 2#+4=0; but there are an endless number of other equations, for example, x 3 2a?-f4=0, # 4 +a? 3 +2=0, and so on without limit. Have we got to make a IMAGINARY NUMBERS 111 new science whenever a new equation ap- pears ? Now, if this were the case, the whole of our preceding investigations, though to some minds they might be amusing, would in truth be of very trifling importance. But the great fact, which has made modern analysis possible, is that, by the aid of this calculus of vectors, every formula which arises can receive its proper interpretation ; and the " unknown " quantity in every equation can be shown to indicate some vector. Thus the science is now complete in itself as far as its fundamental ideas are concerned. It was receiving its final form about the same time as when the steam engine was being perfected, and will remain a great and powerful weapon for the achieve- ment of the victory of thought over things when curious specimens of that machine repose in museums in company with the helmets and breastplates of a slightly earlier epoch. CHAPTER IX COORDINATE GEOMETRY THE methods and ideas of coordinate geo- metry have already been employed in the previous chapters. It is now time for us to consider them more closely for their own sake ; and in doing so we shall strengthen our hold on other ideas to which we have attained. In the present and succeeding chapters we will go back to the idea of the positive and negative real numbers and will ignore the imaginaries which were introduced in the last two chapters. We have been perpetually using the idea that, by taking two axes, XOX' and YOY', in a plane, any point P in that plane can be determined in position by a pair of positive or negative numbers x and t/, where (cf. fig. 13) x is the length OM and y is the length PM. This conception, simple as it looks, is the main idea of the great subject of co- ordinate geometry. Its discovery marks a momentous epoch in the history of mathe- matical thought. It is due (as has been 112 COORDINATE GEOMETRY 113 already said) to the philosopher Descartes, and occurred to him as an important mathe- matical method one morning as he lay in bed. Philosophers, when they have possessed a thorough knowledge of mathematics, have been among those who have enriched the p \y M Y' Fig. 13. science with some of its best ideas. On the other hand it must be said that, with hardly an exception, all the remarks on mathematics made by those philosophers who have pos- sessed but a slight or hasty and late-acquired knowledge of it are entirely worthless, being either trivial or wrong. The fact is a curious one ; since the ultimate ideas of mathematics 114 INTRODUCTION TO MATHEMATICS seem, after all, to be very simple, almost childishly so, and to lie well within the province of philosophical thought. Probably their very simplicity is the cause of error ; we are not used to think about such simple abstract things, and a long training is neces- sary to secure even a partial immunity from error as soon as we diverge from the beaten track of thought. The discovery of coordinate geometry, and also that of projective geometry about the same time, illustrate another fact which is being continually verified in the history of knowledge, namely, that some of the greatest discoveries are to be made among the most well-known topics. By the time that the seventeenth century had arrived, geometry had already been studied for over two thousand years, even if we date its rise with the Greeks. Euclid, taught in the University of Alexandria, being born about 330 B.C. ; and he only systematized and extended the work of a long series of predecessors, some of them men of genius. After him generation after genera- tion of mathematicians laboured at the im- provement of the subject. Nor did the subject suffer from that fatal bar to progress, namely, that its study was confined to a narrow group of men of similar origin and outlook quite the contrary was the case ; by the seventeenth century it had passed COORDINATE GEOMETRY 115 through the minds of Egyptians and Greeks, of Arabs and of Germans. And yet, after all this labour devoted to it through so many ages by such diverse minds its most important secrets were yet to be discovered. No one can have studied even the elements of elementary geometry without feeling the lack of some guiding method. Every proposi- tion has to be proved by a fresh display of in- genuity ; and a science for which this is true lacks the great requisite of scientific thought, namely, method. Now the especial point of coordinate geometry is that for the first time it introduced method. The remote deductions of a mathematical science are not of primary theoretical importance. The science has not been perfected, until it consists in essence of the exhibition of great allied methods by which information, on any desired topic which falls within its scope, can easily be obtained. The growth of a science is not primarily in bulk, but in ideas ; and the more the ideas grow, the fewer are the deductions which it is worth while to write down. Un- fortunately, mathematics is always encum- bered by the repetition in text-books of numberless subsidiary propositions, whose im- portance has been lost by their absorption into the role of particular cases of more general truths and, as we have already in- sisted, generality is the soul of mathematics. 116 INTRODUCTION TO MATHEMATICS Again, coordinate geometry illustrates another feature of mathematics which has already been pointed out, namely, that mathe- matical sciences as they develop dovetail into each other, and share the same ideas in com- mon. It is not too much to say that the various branches of mathematics undergo a perpetual process of generalization, and that as they become generalized, they coalesce. Here again the reason springs from the very nature of the science, its generality, that is to say, from the fact that the science deals with the general truths which apply to all things in virtue of their very existence as things. In this connection the interest of co- ordinate geometry lies in the fact that it relates together geometry, which started as the science of space, and algebra, which has its origin in the science of number. Let us now recall the main ideas of the two sciences, and then see how they are related by Descartes' method of coordinates. Take algebra in the first place. We will not trouble ourselves about the imaginaries and will think merely of the real numbers with posi- tive or negative signs. The fundamental idea is that of any number, the variable number, which is denoted by a letter and not by any definite numeral. We then proceed to the consideration of correlations between vari- ables. For example, if x and y are two vari- ables, we may conceive them as correlated by the equations x+y=I, or by x y=I, or in any one of an indefinite number of other ways. This at once leads to the application of the idea of algebraic form. We think, in fact, of any correlation of some interesting type, thus rising from the initial conception of vari- able numbers to the secondary conception of variable correlations of numbers. Thus we generalize the correlation x+yl, into the correlation ax+by=c. Here a and b and c, being letters, stand for any numbers and are in fact themselves variables. But they are the variables which determine the variable correlation ; and the correlation, when deter- mined, correlates the variable numbers a: and y. Variables, like a, b, and c above, which are used to determine the correlation are called " constants," or parameters. The use of the term " constant " in this connection for what is really a variable may seem at first sight to be odd ; but it is really very natural. For the mathematical investigation is con- cerned with the relation between the variables x and y, after a,b,e are supposed to have been determined. So in a sense, relatively to x and y, the " constants " a, b, and c are con- stants. Thus ax+by=c stands for the general example of a certain algebraic form, that is, for a variable correlation belonging to a certain class. 118 INTRODUCTION TO MATHEMATICS Again we generalize # 2 +t/ 2 =l into by 2 =c, or still further into ax 2 +2hxy +by 2 =c, or, still further, into ax 2 +2hxy +by 2 +2gx Here again we are led to variable correlations which are indicated by their various algebraic forms. Now let us turn to geometry. The name of the science at once recalls to our minds the thought of figures and diagrams exhibiting triangles and rectangles and squares and circles, all in special relations to each other. The study of the simple properties of these figures is the subject matter of elementary geometry, as it is rightly presented to the beginner. Yet a moment's thought will show that this is not the true conception of the subject. It may be right for a child to com- mence his geometrical reasoning on shapes, like triangles and squares, which he has cut out with scissors. What, however, is a tri- angle ? It is a figure marked out and bounded by three bits of three straight lines. Now the boundary of spaces by bits of lines is a ve"ry complicated idea, and not at all one which gives any hope of exhibiting the simple general conceptions which should form the bones of the subject. We want something more simple and more general. It is this obsession with the wrong initial ideas very natural and good ideas for the creation COORDINATE GEOMETRY 119 of first thoughts on the subject which was the cause of the comparative sterility of the study of the science during so many centuries. Coordinate geometry, and Descartes its in- ventor, must have the credit of disclosing the true simple objects for geometrical thought. In the place of a bit of a straight line, let us think of the whole of a straight line throughout its unending length in both direc- tions. This is the sort of general idea from which to start our geometrical investigations. The Greeks never seem to have found any use for this conception which is now funda- mental in all modern geometrical thought. Euclid always contemplates a straight line as drawn between two definite points, and is very careful to mention when it is to be pro- duced beyond this segment. He never thinks of the line as an entity given once for all as a whole. This careful definition and limita- tion, so as to exclude an infinity not immedi- ately apparent to the senses, was very charac- teristic of the Greeks in all their many activities. It is enshrined in the difference between Greek architecture and Gothic archi- tecture, and between the Greek religion and the modern religion. The spire on a Gothic cathedral and the importance of the un- bounded straight line in modern geometry are both emblematic of the transformation of the modern world. 120 INTRODUCTION TO MATHEMATICS The straight line, considered as a whole, is accordingly the root idea from which modern geometry starts. But then other sorts of lines occur to us, and we arrive at the conception of the complete curve which at every point of it exhibits some uniform char- acteristic, just as the straight line exhibits at all points the characteristic of straight- ness. For example, there is the circle which at all points exhibits the characteristic of being at a given distance from its centre, and again there is the ellipse, which is an oval curve, such that the sum of the two distances of any point on it from two fixed points, called its foci, is constant for all points on the curve. It is evident that a circle is merely a particu- lar case of an ellipse when the two foci are superposed in the same point ; for then the sum of the two distances is merely twice the radius of the circle. The ancients knew the properties of the ellipse and the circle and, of course, considered them as wholes. For ex- ample, Euclid never starts with mere seg- ments (i.e., bits) of circles, which are then pro- longed. He always considers the whole circle as described. It is unfortunate that the circle is not the true fundamental line in geometry, so that his defective consideration of the straight line might have been of less consequence. This general idea of a curve which at any COORDINATE GEOMETRY 121 point of it exhibits some uniform property is expressed in geometry by the term " locus." A locus is the curve (or surface, if we do not confine ourselves to a plane) formed by points, all of which possess some given property. To every property in relation to each other which points can have, there corresponds some locus, which consists of all the points possessing the property. In investigating the properties of a locus considered as a whole, we consider any point or points on the locus. Thus in geometry we again meet with the fundamental idea of the variable. Further- more, in classifying loci under such headings as straight lines, circles, ellipses, etc., we again find the idea of form. Accordingly, as in algebra we are concerned with variable numbers, correlations between variable numbers, and the classification of correlations into types by the idea of algebraic form ; so in geometry we are concerned with variable points, variable points satisfying some condition so as form to a locus, and the classification of loci into types by the idea of conditions of the same form. Now, the essence of coordinate geometry is the identification of the algebraic corre- lation with the geometrical locus. The point on a plane is represented in algebra by its two coordinates, x and t/, and the condition satisfied by any point on the locus Is re- 122 INTRODUCTION TO MATHEMATICS presented by the corresponding correlation between x and y. Finally to correlations expressible in some general algebraic form, such as ax+by=c, there correspond loci of some general type, whose geometrical con- ditions are all of the same form. We have thus arrived at a position where we can effect a complete interchange in ideas and results between the two sciences. Each science throws light on the other, and itself gains immeasurably in power. It is im- possible not to feel stirred at the thought of the emotions of men at certain historic moments of adventure and discovery Columbus when he first saw the Western shore, Pizarro when he stared at the Pacific Ocean, Franklin when the electric spark came from the string of his kite, Galileo when he first turned his telescope to the heavens. Such moments are also granted to students in the abstract regions of thought, and high among them must be placed the morning when Descartes lay in bed and inventedfthe method of coordinate geometry. When one has once grasped the idea of co- ordinate geometry, the immediate question which starts to the mind is, What sort of loci correspond to the well-known algebraic forms ? For example, the simplest among the general types of algebraic forms is ax + by=c. The sort of locus which corresponds COORDINATE GEOMETRY 123 to this is a straight line, and conversely to every straight line there corresponds an equa tion of this form. It is fortunate that the simplest among the geometrical loci should correspond to the simplest among the alge- braic forms. Indeed, it is this general corre- spondence of geometrical and algebraic sim- plicity which gives to the whole subject its power. It springs from the fact that the connection between geometry and algebra is not casual and artificial, but deep-seated and essential. The equation which corresponds to a locus is called the equation " of " (or " to ") the locus. Some examples of equations of straight lines will illustrate the subject. ; Fig. 14. 124 INTRODUCTION TO MATHEMATICS Consider y x =0 ; here the a, 6, and c, of the general form have been replaced by 1, 1, and respectively. This line passes through the "origin," 0, in the diagram and bisects the angle XOY. It is the line L'OL of the diagram. The fact that it passes through the origin, 0, is easily seen by observing that the equation is satisfied by putting xQ and t/=0 simultaneously, but and are the co- ordinates of 0. In fact it is easy to generalize and to see by the same method that the equation of any line through the origin is of the form ax-\-by=Q. The loeus of equation /+aj=0 also passes through the origin and bisects the angle X'OY : it is the line L\OL\ of the diagram. Consider y x=\ : the corresponding locus does not pass through the origin. We there- fore seek where it cuts the axes. It must cut the axis of x at some point of coordinates x and 0. But putting t/=0 in the equation, we get x 1; so the coordinates of this point (A) are 1 and 0. Similarly the point (B) where the line cuts the axis OY are and 1. The locus is the line AB in the figure and is parallel to LOU. Similarly y+x=\ is the equation of line AJB of the figure ; and the locus is parallel to L^OL\. It is easy to prove the general theorem that two lines represented by equations of the forms ax-\-by=0 and ax+by=c are parallel. COORDINATE GEOMETRY 125 The group of loci which we next come upon are sufficiently important to deserve a chap- ter to themselves. But before going on to them we will dwell a little longer on the main ideas of the subject. The position of any point P is determined by arbitrarily choosing an origin, 0, two axes, OX and OF, at right-angles, and then by noting its coordinates ac and y, i.e. OM and PM (cf. fig. 13). Also, as we have seen in the last chapter, P can be determined by the " vector " OP, where the idea of the vector includes a determinate direction as well as a determinate length. From an abstract mathematical point of view the idea of an arbitrary origin may appear artificial and clumsy, and similarly for the arbitrarily drawn axes, OX and OY. But in relation to the application of mathematics to the event of the Universe we are here symbolizing with direct simplicity the most fundamental fact respecting the outlook on the world afforded to us by our senses. We each of us refer our sensible perceptions of things to an origin which we call " here " : our location in a particular part of space round which we group the whole Universe is the essential fact of our bodily existence. We can imagine beings who observe all phenomena in all space with an equal eye, unbiassed in favour of any part. With us it is otherwise, a cat at our 126 INTRODUCTION TO MATHEMATICS feet claims more attention than an earth- quake at Cape Horn, or than the destruction of a world in the Milky Way. It is true that in making a common stock of our knowledge with our fellowmen, we have to waive some- thing of the strict egoism of our own indi- vidual " here." We substitute " nearly here " for " here " ; thus we measure miles from the town hall of the nearest town, or from the capital of the country. In measur- ing the earth, men of science will put the origin at the earth's centre ; astronomers even rise to the extreme altruism of putting their origin inside the sun. But, far as this last origin may be, and even if we go further to some convenient point amid the nearer fixed stars, yet, compared to the immeasur- able infinities of space, it remains true that our first procedure in exploring the Universe is to fix upon an origin " nearly here." Again the relation of the coordinates OM and MP (i.e. x and t/) to the vector OP is an instance of the famous parallelogram law, as can easily be seen (cf. fig. 8) by completing the parallelogram OMPN. The idea of the " vector " OP, that is, of a directed magni- tude, is the root-idea of physical science. Any moving body has a certain magnitude of velocity in a certain direction, that is to say, its velocity is a directed magnitude, a vector. Again a force has a certain magni- COORDINATE GEOMETRY 127 tude and has a definite direction. Thus, when in analytical geometry the ideas of the " origin," of " coordinates," and of " vec- tors " are introduced, we are studying the abstract conceptions which correspond to the fundamental facts of the physical world. CHAPTER X CONIC SECTIONS WHEN the Greek geometers had exhausted, as they thought, the more obvious and inter- esting properties of figures made up of straight lines and circles, they turned to the study of other curves ; and, with their almost infallible instinct for hitting upon things worth thinking about, they chiefly (devoted themselves to conic sections, that is, to the curves in which planes would cut the surfaces of circular cones. The man who must have the credit of inventing the study is Menaechmus (born 375 B.C. and died 325 B.C.); he was a pupil of Plato and one of the tutors of Alexander the Great. Alexander, by the by, is a con- spicuous example of the advantages of good tuition, for another of his tutors was the philosopher Aristotle. We may suspect that Alexander found Menaechmus rather a dull teacher, for it is related that he asked for the 128 CONIC SECTIONS 129 proofs to be made shorter. It was to this request that Menaechmus replied : ** In the country there are private and even royal roads, but in geometry there is only one road for all." This reply no doubt was true enough in the sense in which it would have been immediately understood by Alexander. But if Menaechmus thought that his proofs could not be shortened, he was grievously mistaken ; and most modern mathematicians would be horribly bored, if they were com- pelled to study the Greek proofs of the pro- perties of conic sections. Nothing illustrates better the gain in power which is obtained by the introduction of relevant ideas into a science than to observe the progressive shortening of proofs which accompanies the growth of richness in idea. There is a cer- tain type of mathematician who is always rather impatient at delaying over the ideas of a subject : he is anxious at once to get on to the proofs of " important " problems. The history of the science is entirely against him. There are royal roads in science ; but those who first tread them are men of genius and not kings. The way in which conic sections first pre- sented themselves to mathematicians was as follows : think of a cone (cf. fig. 15), whose vertex (or point) is F, standing on a circular base STU, For example, a conical shade to E 130 INTRODUCTION TO MATHEMATICS an electric light is often an example of such a surface. Now let the " generating " lines which pass through V and lie on the surface be all produced backwards; the result is a double cone, and PQR is another circular cross section on the opposite side of V to the cross section STU. The axis of the cone CVC' passes through all the centres of these circles and is perpendicular to their planes, which are parallel to each other. In the diagram the parts of the curves which are supposed to lie behind the plane of the paper are dotted lines, and the parts on the plane or in front of it are continuous lines. Now suppose this double cone is cut by a plane not perpen- dicular to the axis CVC', or at least not necessarily perpendicular to it. Then three cases can arise : (1) The plane may cut the cone in a closed oval curve, such as ABA'B' which lies en- tirely on one of the two half-cones. In this case the plane will not meet the other half-cone at all. Such a curve is called an ellipse ; it is an oval curve. A particular case of such a section of the cone is when the plane is per- pendicular to the axis CFC", then the section, such as STU or PQR, is a circle. IJe^nce a circle is a particular case of the ellipse. ^^ ^ (2) The plane may be parallelled -tangent plane touching the cone along one of its " gen- erating " lines as for example the plane of the CONIC SECTIONS 131 curve DiAiDi' in the diagram is parallel to the tangent plane touching the cone along the generating line VS ; the curve is still confined to one of the half-cones, but it is now not a closed oval curve, it goes on endlessly as long as the generating lines of the half-cone are produced away from the vertex. Such a conic section is called a<sarabola., / (3) The plane may cut botlFtTie half-cones, so that the complete curve consists of two detached portions, or " branches " as they are called, this case is illustrated by the two branches G^^G-i and LiA^L^ which together make up the curve. Neither branch is closed, each of them spreading out endlessly as the two half -cones are prolonged away from the vertex. Such a conic section is called a yperbolay There are accordingly three types of conic sections, namely, ellipses, parabolas, and hyperbolas. It is easy to see that, in a sense, parabolas are limiting cases lying between ellipses and hyperbolas. They form a more special sort and have to satisfy a more par- ticular condition. These three names are apparently due to Apollonius of Perga (born about 260 B.C., and died about 200 B.C.), who wrote a systematic treatise on conic sections which remained the standard work till the sixteenth century. It must at once be apparent how awkward and difficult the investigation of the proper- ties of these curves must have been to the Greek geometers. The curves are plane curves, and yet their investigation involves Fig. 16 the drawing in perspective of a solid figure. Thus in the diagram given above we have practically drawn no subsidiary lines and yet the figure is sufficiently complicated. The CONIC SECTIONS 133 curves are plane curves, and it seems obvious that we should be able to define them without going beyond the plane into a solid figure. At the same time, just as in the " solid " Fig. 17 definition there is one uniform method of definition namely, the section of a cone by 134 INTRODUCTION TO MATHEMATICS a plane which yields three cases, so in any " plane " definition there also should be one uniform method of procedure which falls into three cases. Their shapes when drawn on their planes are those of the curved lines in the three figures 16, 17, and 18. The points A and A' in the figures are called Fig. 18 the vertices and the line AA' the major axis. It will be noted that a parabola (cf. fig. 17) has only one vertex. Apollonius proved * that the ratio of PM 2 to AM.MA ' (i.e. -^^- \ AM.MA remains constant both for the ellipse and the hyperbola (figs. 16 and 18), and that the ratio * Cf. Ball, loc. cit., for this account of Apollonius and Pappus. CONIC SECTIONS 135 of PM 2 to AM is constant for the parabola of fig. 17 ; and he bases most of his work on this fact. We are evidently advancing towards the desired uniform definition which does not go out of the plane ; but have not yet quite attained to uniformity. In the diagrams 16 and 18, two points, S and S', will be seen marked, and in diagram 17 one point, S. These are the foci of the curves, and are points of the greatest importance. Apollonius knew that for an ellipse the sum of SP and S'P (i.e. SP+S'P) is constant as P moves on the curve, and is equal to A A'. Similarly for a hyperbola the difference S'P S'P is constant, and equal to A A' when P is on one branch, and the difference SP' S'P' is constant and equal to A A' when P' is on the other branch. But no corresponding point seemed to exist for the parabola. Finally 500 years later the last great Greek geometer, Pappus of Alexandria, discovered the final secret which completed this line of thought. In the diagrams 16 and 18 will be seen two lines, XN and X'N', and in diagram 17 the single line, XN. These are the direc- trices of the curves, two each for the ellipse and the hyperbola, and one for the parabola. Each directrix corresponds to its nearer focus. The characteristic property of a focus, S, and its corresponding directrix, XN, for any one of the three types of curve, is that the ratio 136 INTRODUCTION TO MATHEMATICS (rrr>\ i.e. pr) is constant, where PN is the perpendicular on the directrix fromP, and P is any point on the curve. Here we have finally found the desired property of the curves which does not require us to leave the plane, and is stated uniformly for all OIT> three curves. For ellipses the ratio* ^^ is less PN than 1, for parabolas it is equal to 1, and for hyperbolas it is greater than 1. When Pappus had finished his investiga- tions, he must have felt that, apart from minor extensions, the subject was practically exhausted ; and if he could have foreseen the history of science for more than a thousand years, it would have confirmed his belief. Yet in truth the really fruitful ideas in con- nection with this branch of mathematics had not yet been even touched on, and no one had guessed their supremely important ap- plications in nature. No more impressive warning can be given to those who would confine knowledge and research to what is apparently useful, than the reflection that conic sections were studied for eighteen hun- dred years merely as an abstract science, without a thought of any utility other than to satisfy the craving for knowledge on the part of mathematicians, and that then at the end of this long period of abstract study, they * Cf. Note B, p. 250. CONIC SECTIONS 137 were found to be the necessary key with which to attain the knowledge of one of the most important laws of nature. Meanwhile the entirely distinct study of astronomy had been going forward. The great Greek astronomer Ptolemy (died 168 A.D.) published his standard treatise on the subject in the University of Alexandria, ex- plaining the apparent motions among the fixed stars of the sun and planets by the con- ception of the earth at rest and the sun and the planets circling round it. During the next thirteen hundred years the number and the accuracy of the astronomical observa- tions increased, with the result that the de- scription of the motions of the planets on Ptolemy's hypothesis had to be made more and more complicated. Copernicus (born 1473 A.D. and died 1543 A.D.) pointed out that the motions of these heavenly bodies could be explained in a simpler manner if the sun were supposed to rest, and the earth and planets were conceived as moving round it. However, he still thought of these motions as essentially circular, though modified by a set of small corrections arbitrarily superimposed on the primary circular motions. So the matter stood when Kepler was born at Stutt- gart in Germany in 1571 A.D. There were two sciences, that of the geometry of conic sections and that of astronomy, both of which 138 INTRODUCTION TO MATHEMATICS had been studied from a remote antiquity without a suspicion of any connection be- tween the two. Kepler was an astronomer, but he was also an able geometer, and on the subject of conic sections had arrived at ideas in advance of his time He is only one of many examples of the falsity of the idea that success in scientific research demands an ex- clusive absorption in one narrow line of study. Novel ideas are more apt to spring from an unusual assortment of knowledge not necessarily from vast knowledge, but from a thorough conception of the methods and ideas of distinct lines of thought. It will be re- membered that Charles Darwin was helped to arrive at his conception of the law of evolution by reading Malthus' famous Essay on Population, a work dealing with a dif- ferent subject at least, as it was then thought. Kepler enunciated three laws of planetary motion, the first two in 1609, and the third ten years later. They are as follows : (1) The orbits of the planets are ellipses, the sun being in the focus. (2) As a planet moves in its orbit, the radius vector from the sun to the planet sweeps out equal areas in equal times. (3) The squares of the periodic times of the several planets are proportional to the cubes of their major axes. CONIC SECTIONS 139 These laws proved to be only a stage to- wards a more fundamental development of ideas. Newton (born 1642 A.D. and died 1727 A.D.) conceived the idea of universal gravitation, namely, that any two pieces of \ matter attract each other with a force pro- portional to the product of their masses and , inversely proportional to the square of their distance from each other. This sweeping general law, coupled with the three laws of motion which he put into their final general shape, proved adequate to explain all astro- nomical phenomena, including Kepler's laws, and has formed the basis of modern physics. Among other things he proved that comets might move in very elongated ellipses, or in parabolas, or in hyperbolas, which are nearly parabolas. The comets which return such\ as Halley's comet must, of course, move in I ellipses. But the essential step in the proof of the law of gravitation, and even in the sug- gestion of its initial conception, was the veri- fication of Kepler's laws connecting the motions of the planets with the theory of conic sections. From the seventeenth century onwards the abstract theory of the curves has shared in the double renaissance of geometry due to the introduction of coordinate geometry and of projective geometry. In pEpjeetuta-gee* metry the fundamental ideas cluster round 140 INTRODUCTION TO MATHEMATICS the consideration of sets (or pencils, as they are called) of lines passing through a common point (the vertex of the " pencil "). Now (cf. fig. 19) if A, B, C, Z), be any four fixed points on a conic section and P be a variable point on the curve, the pencil of lines PA, Fig. 19. PB, PC, and PD, has a special property, known as the constancy of its cross ratio. It will suffice here to say that cross ratio is a fundamental idea in projective geometry. For projective geometry this is really the de- finition of the curves, or some analogous pro- perty which is really equivalent to it, It CONIC SECTIONS 141 will be seen how far in the course of ages of study we have drifted away from the old original idea of the sections of a circular cone. We know now that the Greeks had got hold of a minor property of comparatively slight importance ; though by some divine good fortune the curves themselves deserved all the attention which was paid to them. This unimportance of the '* section " idea is now marked in ordinary mathematical phrase- ology by dropping the word from their names. As often as not, they are now named merely " conies " instead of " conic sections." Finally, we come back to the point at which we left coordinate geometry in the last chapter. We had asked what was the type of loci corresponding to the general algebraic form ax-\-by=c, and had found that it was the class of straight lines in the plane. We had seen that every straight line possesses an equation of this form, and that every equation of this form corresponds to a straight line. We now wish to go on to the next general type of algebraic forms. This is evidently to be obtained by introducing terms involv- ing x 2 and xy and y 2 . Thus the new general form must be written x +2fy +c =0 What does this represent ? The answer is 142 INTRODUCTION TO MATHEMATICS that (when it represents any locus) it always re- presents a conic section, and, furthermore, that the equation of every conic section can always be put into this shape. The discrimi- nation of the particular sorts of conies as given by this form of equation is very easy. It en- tirely depends upon the consideration of ab h 2 , where a, b, and h, are the " constants " as written above. If abh 2 is a positive number, the curve is an ellipse ; if abh 2 ~0, the curve is a parabola : and if abh 2 is a negative number, the curve is a hyperbola. For example, put a =6=1, h=g=f=Q, c= 4. We then get the equation x 2 -\-y 2 4 =0. It is easy to prove that this is the equa- tion of a circle, whose centre is at the origin, and radius is 2 units of length. Now abh 2 becomes 1x1 O 2 , that is, 1, and is therefore positive. Hence the circle is a particular case of an ellipse, as it ought to be. Genera- lising, the equation of any circle can be put into the form a(x 2 -\-y 2 } -\-2gcc -\-2fy+c=0. Hence abh 2 becomes a 2 0, that is, a 2 , which is necessarily positive. Accordingly all circles satisfy the condition for ellipses. The general form of the equation of a para- bola is so that the terms of the second degree, as CONIC SECTIONS 143 they are called, can be written as a perfect square. For squaring out, we get d z x 2 +2dexy +e 2 y 2 +2gx+2fy +c ; so that by comparison a=d 2 , h=de, b=e z , and therefore ab h 2 =d 2 e 2 (de) 2 =0. Hence the necessary condition is automatically satis- fied. The equation 2xy 4=0, where a =b =g=/=0, h=l, c= 4, represents a hyper- bola. For the condition abh 2 becomes I 2 , that is, 1, which is negative. The limitation, introduced by saying that, when the general equation represents any locus, it represents a conic section, is necessary, be- cause some particular cases of the general equation represent no real locus. For ex- ample x 2 -\-y 2 +I=Q can be satisfied by no real values of x and y. It is usual to say that the locus is now one composed of imaginary points. But this idea of imaginary points in geometry is really one of great complexity, which we will not now enter into. Some exceptional cases are included in the general form of the equation which may not be immediately recognized as conic sections. By properly choosing the constants the equa- tion can be made to represent two straight lines. Now two intersecting straight lines may fairly be said to come under the Greek idea of a conic section. For, by referring to 144 INTRODUCTION TO MATHEMATICS the picture of the double cone above, it will be seen that some planes through the vertex, F, will cut the cone in a pair of straight lines intersecting at V. The case of two parallel straight lines can be included by considering a circular cylinder as a particular case of a cone. Then a plane, which cuts it and is parallel to its axis, will cut it in two parallel straight lines. Anyhow, whether or no the ancient Greek would have allowed these special cases to be called conic sections, they are certainly included among the curves re- presented by the general algebraic form of the second degree. This fact is worth noting ; for it is characteristic of modern mathematics to include among general forms all sorts of particular cases which would formerly have received special treatment. This is due to its pursuit of generality. CHAPTER XI FUNCTIONS THE mathematical use of the term function has been adopted also in common life. For example, " His temper is a function of his digestion," uses the term exactly in this mathematical sense. It means that a rule can be assigned which will tell you what his temper will be when you know how his digestion is working. Thus the idea of a " function " is simple enough, we only have to see how it is applied in mathematics to variable numbers. Let us think first of some concrete examples : If a train has been travel- ling at the rate of twenty miles per hour, the distance (s miles) gone after any number of hours, say t, is given by s=20xt; and s is called a function of t. Also 20 xt is thTTunc^ tion of t with which s is identical. If John is one year older than Thomas, then, when Thomas is at any age of x years, John's age (y years) is given by y=x+I ; and y is a function of #, namely, is the function x+\. In these examples t and x are called the 145 146 INTRODUCTION TO MATHEMATICS " arguments " of the functions in which they appear. Thus t is the argument of the func- tion 20 xt, and x is the argument of the func- tion #-{-1. If s=20xt, and t/=a?+l, tAen ^ and y are called the "values " of the functions 20 xt and a?-f-l respectively. Coming now to the general case, we can define a function in mathematics as a corre- lation between two variable numbers, called respectively the argument and the value of the function, such that whatever value be assigned to the " argument of the function " the " value of the function " is definitely (i.e. uniquely) determined. The converse is not necessarily true, namely, that when the value of the function is determined the argument is also uniquely determined. Other functions of the argument x are yx 2 , t/=2a? 2 -|-3#+l, y=x, y=log x, y=sin x. The last two functions of this group will be readily recognizable by those who understand a little algebra and trigonometry. It is not worth while to delay now for their explana- tion, as they are merely quoted for the sake of example. Up to this point, though we have defined what we mean by a function in general, we have only mentioned a series of special func- tions. But mathematics, true to its general methods of procedure, symbolizes the general idea of any function. It does this by writing FUNCTIONS 147 f\x), /(#), g(x), <f>(x), etc., for any function of x, where the argument x is placed in a bracket, and some letter like F, /, g, <f>, etc., is prefixed to the bracket to stand for the function. This notation has its defects. Thus it obvi- ously clashes with the convention that the single letters are to represent variable num- bers ; since here F, /, g, <, etc., prefixed to a bracket stand for variable functions. It would be easy to give examples in which we can only trust to common sense and the con- text to see what is meant. One way of evading the confusion is by using Greek letters (e.g. <f> as above) for functions ; an- other way is to keep to / and F (the initial letter of function) for the functional letter, and, if other variable functions have to be symbolized, to take an adjacent letter like g. With these explanations and cautions, we write yf(x) t to denote that y is the value of some undetermined function of the argument x ; where f(x) may stand for anything such as a?+l, x 2 2#4-l, sin x, log x, or merely for x itself. The essential point is that when x is given, then y is thereby definitely deter- mined. It is important to be quite clear as to the generality of this idea. Thus in y = f(x), we may determine, if we choose, f(x) to mean that when x is an integer, f(x) is zero, and when x has any other value, f(x) is 1. Accordingly, putting t/ =/(#), with this choice 148 INTRODUCTION TO MATHEMATICS for the meaning of /, y is either or 1 accord- ing as the value of x is integral or otherwise. Thus /(1)=0, /(2)=0, /()=!, /(V2)=l, and so on. This choice for the meaning of f(x) gives a perfectly good function of the argu- ment x according to the general definition of a function. A function, which after all is only a sort of correlation between two variables, is re- presented like other correlations by a graph, that is in effect by the methods of coordinate geometry. For example, fig. 2 in Chapter II. is the graph of the function - where v is the argument and p the value of the function. In this case the graph is only drawn for positive values of v, which are the only values possessing any meaning for the physical ap- plication considered in that chapter. Again in fig. 14 of Chapter IX. the whole length of the line AB, unlimited in both directions, is the graph of the function aj+l, where x is the argument and y is the value of the function ; and in the same figure the unlimited line AiB is the graph of the function 1 x, and the line LOL' is the graph of the function x, x being the argument and y the value of the function. These functions, which are expressed by simple algebraic formulae, are adapted for re- presentation by graphs. But for some func- FUNCTIONS 149 tions this representation would be very misleading without a detailed explanation, or might even be impossible. Thus, consider the function mentioned above, which has the value 1 for all values of its argument x, except those which are integral, e.g. except for #=0, o:=l, #=2, etc., when it has the value 0. Its appearance on a graph would be that of the straight line ABA' drawn parallel to the A' I B, B 2 B 3 B 4 BS 12345 Fig. 20. axis XOX' at a distance from it of 1 unit of length. But the points, B, Ci, C%, 3, C^, etc., corresponding to the values 0, 1, 2, 3, 4, etc., of the argument x, are to be omitted, and in- stead of them the points 0, B\, B%, B& B, etc., on the axis OX, are to be taken. It is easy to find functions for which the graphical re- presentation is not only inconvenient but impossible. Functions which do not lend themselves to graphs are important in the 150 INTRODUCTION TO MATHEMATICS higher mathematics, but we need not concern ourselves further about them here. * The most important division between func- tions is that between continuous and discon- tinuous functions. A function is continuous when its value only alters gradually for gradual alterations of the argument, and is discontinuous when it can alter its value by sudden jumps. Thus the two functions #+1 and 1x, whose graphs are depicted as straight lines in fig. 14 of Chapter IX., are con- tinuous functions, and so is the function -, depicted in Chapter II., if we only think of positive values of v. But the function de- picted in fig. 20 of this chapter is discontinuous since at the values o?=l, x=2, etc., of its argument, its value gives sudden jumps. Let us think of some examples of functions presented to us in nature, so as to get into our heads the real bearing of continuity and discontinuity. Consider a train in its journey along a railway line, say from Euston Station, the terminus in London of the London and North- Western Railway. Along the line in order lie the stations of Bletchley and Rugby. Let t be the number of hours which the train has been on its journey from Euston, and s be the number of miles passed over. Then * is a function of t, i.e. is the variable value corresponding to the variable argument t. FUNCTIONS 151 If we know the circumstances of the train's run, we know s as soon as any special value of t is given. Now, miracles apart, we may confidently assume that s is a continuous function of t. It is impossible to allow for the contingency that we can trace the train continuously from Euston to Bletchley, and that then, without any intervening time, how- ever short, it should appear at Rugby. The idea is too fantastic to enter into our calcula- tion : it contemplates possibilities not to be found outside the Arabian Nights ; and even in those tales sheer discontinuity of motion hardly enters into the imagination, they do not dare to tax our credulity with anything more than very unusual speed. But unusual speed is no contradiction to the great law of continuity of motion which appears to hold in nature. Thus light moves at the rate of about 190,000 miles per second and comes to us from the sun in seven or eight minutes ; but, in spite of this speed, its distance travelled is always a continuous function of the time. It is not quite so obvious to us that the velocity of a body is invariably a continuous function of the time. Consider the train at any time t : it is moving with some definite velocity, say v miles per hour, where v is zero when the train is at rest in a station and is negative when the train is backing. Now we readily allow that v cannot change its 152 INTRODUCTION TO MATHEMATICS value suddenly for a big, heavy train. The train certainly cannot be running at forty miles per hour from 11.45 a.m. up to noon, and then suddenly, without any lapse of time, commence running at 50 miles per hour. We at once admit that the change of velocity will be a gradual process. But how about sudden blows of adequate magnitude ? Sup- pose two trains collide ; or, to take smaller objects, suppose a man kicks a football. It certainly appears to our sense as though the football began suddenly to move. Thus, in the case of velocity our senses do not revolt at the idea of its being a discontinuous func- tion of the time, as they did at the idea of the train being instantaneously transported from Bletchley to Rugby. As a matter of fact, if the laws of motion, with their conception of mass, are true, there is no such thing as discontinuous velocity in nature. Anything that appears to our senses as discontinuous change of velocity must, according to them, be considered to be a case of gradual change which is too quick to be perceptible to us. It would be rash, however, to rush into the generalization that no discontinuous functions are presented to us in nature. A man who, trusting that the mean height of the land above sea-level between London and Paris was a continuous function of the distance from London, walked at night on Shakes- FUNCTIONS 153 peare's Cliff by Dover in contemplation of the Milky Way, would be dead before he had had time to rearrange his ideas as to the necessity of caution in scientific conclusions. It is very easy to find a discontinuous function, even if we confine ourselves to the simplest of the algebraic formulae. For ex- ample, take the function y=-, which we x have already considered in the form p=-> where v was confined to positive values. But 154 INTRODUCTION TO MATHEMATICS now let x have any value, positive or negative. The graph of the function is exhibited in fig. 21. Suppose x to change continuously from a large negative value through a numerically decreasing set of negative values up to 0, and thence through the series of increasing posi- tive values. Accordingly, if a moving point, M, represents x on XOX', M starts at the extreme left of the axis XOX' and succes- sively moves through MI, MZ, MZ, M*, etc. The corresponding points on the function are PI, P2, PS, P-i, etc. It is easy to see that there is a point of discontinuity at #=0, i.e. at the origin O. For the value of the function on the negative (left) side of the origin be- comes endlessly great, but negative, and the function reappears on the positive (right) side as endlessly great but positive. Hen.ce, however small we take the length M 2 M 3 , there is a finite jump between the values of the function at M% and M 3 . Indeed, this case has the peculiarity that the smaller we take the length between M% and MS, so long as they enclose the origin, the bigger is the jump in value of the function between them. This graph brings out, what is also apparent in fig. 20 of this chapter, that for many functions the discontinuities only occur at isolated points, so that by restricting the values of the argument we obtain a continuous function for these remaining values. Thus it is evident FUNCTIONS 155 from fig. 21 that in y =-, if we keep to positive x values only and exclude the origin, we obtain a continuous function. Similarly the same function, if we keep to negative values only, excluding the origin, is continuous. Again the function which is graphed in fig. 20 is con- tinuous between B and Ci, and between C\ and 2, and between C% and C& and so on, always in each case excluding the end points. It is, however, easy to find functions such thaT? their discontinuities occur at all points. For j example, consider a function /(#), such thaf^ when x is any fractional number /(#)=!, and when x is any incommensurable number /(#) =2. This function is discontinuous at all points. Finally, we will look a little more closely at the definition of continuity given above. We have said that a function is continuous when its value only alters gradually for gradual alterations of the argument, and is discontinuous when it can alter its value by sudden jumps. This is exactly the sort of definition which satisfied our mathematical i forefathers and no longer satisfies modern .mathematicians. It is worth while to spend ji some time over it ; for when we understand the modern objections to it, we shall have :gone a long way towards the understanding of the spirit of modern mathematics. The 156 INTRODUCTION TO MATHEMATICS whole difference between the older and the newer mathematics lies in the fact that vague half -metaphorical terms like " gradually " are no longer tolerated in its exact statements. Modern mathematics will only admit state- ments and definitions and arguments which exclusively employ the few simple ideas about number and magnitude and variables on which the science is founded. Of two num- bers one can be greater or less than the other ; and one can be such and such a multi- ple of the other ; but there is no relation of " graduality " between two numbers, and hence the term is inadmissible. Now this may seem at first sight to be great pedantry. To this charge there are two answers. In the first place, during the first half of the nineteenth century it was found by some great mathematicians, especially Abel in Sweden, and Weierstrass in Germany, that large parts of mathematics as enunciated in the old happy-go-lucky manner were simply wrong. Macaulay in his essay on Bacon contrasts the certainty of mathematics with the uncertainty of philosophy ; and by way of a rhetorical example he says, " There has been no reaction against Taylor's theorem." He could not have chosen a worse example. For, without having made an examination of English text-books on mathematics contem- porary with the publication of this essay, the FUNCTIONS 157 assumption is a fairly safe one that Taylor's theorem was enunciated and proved wrongly in every one of them. Accordingly, the anxious precision of modern mathematics is necessary for accuracy. In the second place it is necessary for research. It makes for clearness of thought, and thence for boldness of thought and for fertility in trying new combinations of ideas. When the initial statements are vague and slipshod, at every subsequent stage of thought common sense has to step in to limit applications and to explain meanings. Now in creative thought common sense is a bad master. Its sole criterion for judgment is that the new ideas shall look like the old ones. In other words it can only act by suppressing originality. In working our way towards the precise definition of continuity (as applied to func- tions) let us consider more closely the state- ment that there is no relation of " graduality " between numbers. It may be asked, Cannot one number be only slightly greater than another number, or in other words, cannot the difference between the two numbers be small ? The whole point is that in the ab- stract, apart from some arbitrarily assumed application, there is no such thing as a great or a small number. A million miles is a small number of miles for an astronomer investigating the fixed stars, but a million 158 INTRODUCTION TO MATHEMATICS pounds is a large yearly income. Again, one- quarter is a large fraction of one's income to give away in charity, but is a small fraction of it to retain for private use. Examples can be accumulated indefinitely to show that great or small in any absolute sense have no abstract application to numbers. We can say'bf two numbers that one is greater or smaller than another, but not without speci- fication of particular circumstances that any one number is great or small. Our task therefore is to define continuity without any mention of a " small " or " gradual " change in value of the function. In order to do this we will give names to some ideas, which will also be useful when we come to consider limits and the differential calculus. An " interval " of values of the argument a? of a function /(#) is all the values lying between some two values of the argument. For example, the interval between x=l and #=2 consists of all the values which x can take lying between 1 and 2, i.e. it consists of all the real numbers between 1 and 2. But the bounding numbers of an interval need not be integers. An interval of values of the argument contains a number a, when a is a member of the interval. For example, the interval between 1 and 2 contains f , f , , and so on. FUNCTIONS 159 A set of numbers approximates to a num- ber a within a standard k, when the numerical difference between a and every number of the set is less than k. Here k is the " standard of approximation." Thus the set of num- bers 3, 4, 6, 8, approximates to the number 5 within the standard 4. In this case the standard 4 is not the smallest which could have been chosen, the set also approximates to 5 within any of the standards 3-1 or 3 -01 or 3-001. Again, the numbers, 3-1, 3-141, 3-1415, 3-14159 approximate to 3-13102 with- in the standard -032, and also within the smaller standard -03103. These two ideas of an interval and of approximation to a number within a standard are easy enough ; their only difficulty is that they look rather trivial. But when combined with the next idea, that of the " neighbour- hood " of a number, they form the foundation of modern mathematical reasoning. What do we mean by saying that something is true for a function f(x) in the neighbourhood of the value a of the argument x ? It is this fundamental notion which we have now got to make precise. The values of a function f(x) are said to possess a characteristic in the " neighbour- hood of a " when some interval can be found, which (i) contains the number a not as an end-point, and (ii) is such that every value 160 INTRODUCTION TO MATHEMATICS of the function for arguments, other than a, lying within that interval possesses the char- acteristic. The value /(a) of the function for the argument a may or may not possess the characteristic. Nothing is decided on this point by statements about the neighbourhood of a. For example, suppose we take the particu- lar function x 2 . Now in the neighbourhood of 2, the values of x 2 are less than 5. For we can find an interval, e.g. from 1 to 2 f l, which (i) contains 2 not as an end-point, and (ii) is such that, for values of x lying within it, x 2 is less than 5. Now, combining the preceding ideas we know what is meant by saying that in the neighbourhood of a the function f(x) approxi- mates to c within the standard k. It means that some interval can be found which (i) includes a not as an end-point, and (ii) is such that all values of /(#), where x lies in the inter- val and is nota, differ fromc by less than k. For example, in the neighbourhood of 2, the func- tion i/x approximates to 1-41425 within the standard '0001. This is true because the square root of 1-99996164 is 1-4142 and the square root of 2*00024449 is 1-4143 ; hence for values of x lying in the interval 1-99996164 to 2-00024449, which contains 2 not as an end-point, the values of the function <x all lie between 1-4142 and 1-4143, and FUNCTIONS 161 they therefore all differ from 1 '41425 by less than '0001. In this case we can, if we like, fix a smaller standard of approximation,^ namely '000051 or -0000501. Again, to take another example, in the neighbourhood of 2 the function x 2 approximates to 4 within the standard -5. For (1'9) 2 =3'61 and (2 1)2 = 4*41, and thus the required interval 1*9 to 2'1, containing 2 not as an end-point, has been found. This example brings out the fact that statements about a function f(x) in the neighbourhood of a number a are distinct from statements about the value of f(x) when x =a. The production of an interval, through- out which the statement is true, is required. Thus the mere fact that 2 2 =4 does not by itself justify us in saying that in the neigh- bourhood of 2 the function x 2 is equal to 4. This statement would be untrue, because no interval can be produced with the required property. Also, the fact that 2 2 =4 does not by itself justify us in saying that in the neighbourhood of 2 the function x 2 approxi- mates to 4 within the standard '5 ; although as a matter of fact, the statement has just been proved to be true. If we understand the preceding ideas, we understand the foundations of modern mathematics. We shall recur to analogous ideas in the chapter on Series, and again in the chapter on the Differential Calculus. 162 INTRODUCTION TO MATHEMATICS Meanwhile, we are now prepared to define " continuous functions." A function /(a?) is " continuous " at a value a of its argu- ment, when in the neighbourhood of a its values approximate to /(a) (i.e. to its value at a) within every standard of ap- proximation. This means that, whatever standard k be chosen, in the neighbourhood of a j(x) ap- proximates to /(a) within the standard k. For example, x 2 is continuous at the value 2 of its argument, #, because however k be chosen we can always find an interval, which (i) contains 2 not as an end-point, and (ii) is such that the values of x 2 for arguments lying within it approximate to 4 (i.e. 2 2 ) within the standard k. Thus, suppose we choose the standard -1 ; now (1'999) 2 =3'996001, and (2 -01 ) 2 = 4*0401, and both these numbers differ from 4 by less than *1. Hence, within the interval 1-999 to 2 -01 the values of x 2 approximate to 4 within the standard !. Similarly an interval can be produced for any other standard which we like to try. Take the example of the railway train. Its velocity is continuous as it passes the signal box, if whatever velocity you like to assign (say one-millionth of a mile per hour) an in- terval of time can be found extending before and after the instant of passing, such that at all instants within it the train's velocity FUNCTIONS 163 differs from that with which the train passed the box by less than one-millionth of a mile per hour ; and the same is true whatever other velocity be mentioned in the place of one-millionth of a mile per hour. CHAPTER XII PERIODICITY IN NATURE THE whole life of Nature is dominated by the existence of periodic events, that is, by the existence of successive events so analogous to each other that, without any straining of language, they may be termed recurrences of the same event. The rotation of the earth produces the successive days. It is true that each day is different from the preceding days, however abstractly we define the meaning of a day, so as to exclude casual phenomena. But with a sufficiently abstract definition of a day, the distinction in properties between two days becomes faint and remote from practical interest ; and each day may then be conceived as a recurrence of the phenome- non of one rotation of the earth. Again the path of the earth round the sun leads to the yearly recurrence of the seasons, and imposes another periodicity on all the operations of nature. Another less fundamental perio- dicity is provided by the phases of the moon. In modern civilized life, with its artificial light, these phases are of slight importance, but in 164 PERIODICITY IN NATURE 165 ancient times, in climates where the days are burning and the skies clear, human life was apparently largelyinfluencedby the existenceof moonlight. Accordingly our divisions into weeks and months, with their religious associa- tions, have spread over the European races from Syria and Mesopotamia, though independent observances following the moon's phases are found amongst most nations. It is, however, through the tides, and not through its phases of light and darkness, that the moon's perio- dicity has chiefly influenced the history of the earth. Our bodily life is essentially periodic. It is dominated by the beatings of the heart, and the recurrence of breathing. The presupposition of periodicity is indeed fundamental to our very conception of life. We cannot imagine a course of nature in which, as events progressed, we should be unable to say : " This has happened before." The whole conception of experience as a guide to conduct would be absent. Men would always find themselves in new situations possessing no substratum of identity with anything in past history. The very means of measuring time as a quantity would be absent. Events might still be recognized as occurring in a series, so that some were earlier and others later. But we now go beyond this bare recognition. We can not only say that 16 INTRODUCTION TO MATHEMATICS three events, A, B, C, occurred in this order, so that A came before B, and B before C ; but also we can say that the length of time between the occurrences of A and B was twice as long as that between B and C. Now, quantity of time is essentially dependent on observing the number of natural recurrences which have intervened. We may say that the length of time between A and B was so many days, or so many months, or so many years, according to the type of recur- rence to which we wish to appeal. Indeed, at the beginning of civilization, these three modes of measuring time were really distinct. It has been one of the first tasks of science among civilized or semi-civilized nations, to fuse them into one coherent measure. The full extent of this task must be grasped. It is necessary to determine, not merely what number of days (e.g. 365 '25 . . .) go to some one year, but also previously to determine that the same number of days do go to the suc- cessive years. We can imagine a world in which periodicities exist, but such that no two are coherent. In some years there might be 200 days and in others 350. The determina- tion of the broad general consistency of the more important periodicities was the first step in natural science. This consistency arises from no abstract intuitive law of thought ; it is merely an observed fact of nature PERIODICITY IN NATURE 167 guaranteed by experience. Indeed, so far is it from being a necessary law, that it is not even exactly true There are divergencies in every case. For some instances these diver- gencies are easily observed and are therefore immediately apparent. In other cases it re- quires the most refined observations and astronomical accuracy to make them appar- ent. Broadly speaking, all recurrences de- pending on living beings, such as the beatings of the heart, are subject in comparison with other recurrences to rapid variations. The great stable obvious recurrences stable in the sense of mutually agreeing with great accuracy are those depending on the motion of the earth as a whole, and on similar motions of the heavenly bodies. We therefore assume that these astronomi- cal recurrences mark out equal intervals of time. But how are we to deal with their discrepancies which the refined observations of astronomy detect ? Apparently we are reduced to the arbitrary assumption that one or other of these sets of phenomena marks out equal times e.g. that either all days are of equal length, or that all years are of equal length. This is not so : some assumptions must be made, but the assumption which underlies the whole procedure of the astrono- mers in determining the measure of time is that the laws of motion are exactly verified. Before explaining how this is done, it is in- teresting to observe that this relegation of the determination of the measure of time to the astronomers arises (as has been said) from the stable consistency of the recurrences with which they deal. If such a superior con- sistency had been noted among the recur- rences characteristic of the human body, we should naturally have looked to the doctors of medicine for the regulation of our clocks. In considering how the laws of motion come into the matter, note that two incon- sistent modes of measuring time will yield different variations of velocity to the same body. For example, suppose we define an hour as one twenty-fourth of a day, and take the case of a train running uniformly for two hours at the rate of twenty miles per hour. Now take a grossly inconsistent measure of time, and suppose that it makes the first hour to be twice as long as the second hour. Then, according to this other measure of duration, the time of the train's run is divided into two parts, during each of which it has tra- versed the same distance, namely, twenty miles ; but the duration of the first part is twice as long as that of the second part. Hence the velocity of the train has not been uniform, and on the average the velocity during the second period is twice that during the first period. Thus the question as to PERIODICITY IN NATURE 169 whether the train has been running uniformly or not entirely depends on the standard of time which we adopt. Now, for all ordinary purposes of life on the earth, the various astronomical recurrences may be looked on as absolutely consistent ; and, furthermore assuming their consistency, and thereby assuming the velocities and changes of velocities possessed by bodies, we find that the laws of motion, which have been considered above, are almost exactly verified. But only almost exactly when we come to some of the astronomical phenomena. We find, however, that by assuming slightly different velocities for the rotations and motions of the planets and stars, the laws would be exactly verified. This assumption is then made ; and we have, in fact thereby, adopted a measure of time, which is indeed defined by reference to the astronomical phenomena, but not so as to be consistent with the uniformity of any one of them. But the broad fact remains that the uniform flow of time on which so much is based, is itself dependent on the observation of periodic events. Even phenomena, which on the surface seem casual and exceptional, or, on the other hand, maintain themselves with a uniform persistency, may be due to the remote influ- ence of periodicity. Take for example, the 170 INTRODUCTION TO MATHEMATICS principle of resonance. Resonance arises when two sets of connected circumstances have the same periodicities. It is a dynami- cal law that the small vibrations of all bodies when left to themselves take place in definite times characteristic of the body. Thus a pendulum with a small swing always vibrates in some definite time, characteristic of its shape and distribution of weight and length. A more complicated body may have many ways of vibrating ; but each of its modes of vibration will have its own peculiar " period." Those periods of vibration of a body are called its ** free " periods. Thus a pendulum has but one period of vibration, while a suspension bridge will have many. We get a musical instrument, like a violin string, when the periods of vibration are all simple submultiples of the longest ; i.e. if t seconds be the longest period, the others are \t, \t, and so on, where any of these smaller periods may be absent. Now, suppose we excite the vibrations of a body by a cause which is itself periodic; then, if the period of the cause is very nearly that of one of the periods of the body, that mode of vibration of the body is very violently excited ; even although the magnitude of the exciting cause is small. This phenomenon is called " resonance." The general reason is easy to understand. Any one wanting to upset a rocking stone will push " in tune " PERIODICITY IN NATURE 171 with the oscillations of the stone, so as always to secure a favourable moment for a push. If the pushes are out of tune, some increase the oscillations, but others check them. But when they are in tune, after a time all the pushes are favourable. The word " reson- ance " comes from considerations of sound : but the phenomenon extends far beyond the region of sound. The laws of absorption and emission of light depend on it, the " tuning " of receivers for wireless telegraphy, the com- parative importance of the influences of planets on each other's motion, the danger to a suspension bridge as troops march over it in step, and the excessive vibration of some ships under the rhythmical beat of their machinery at certain speeds. This coinci- dence of periodicities may produce steady phenomena when there is a constant associ- ation of the two periodic events, or it may produce violent and sudden outbursts when the association is fortuitous and temporary. Again, the characteristic and constant periods of vibration mentioned above are the underlying causes of what appear to us as steady excitements of our senses. We work for hours in a steady light, or we listen to a steady unvarying sound. But, if modern science be correct, this steadiness has no counterpart in nature. The steady light is due to the impact on the eye of a countless 172 INTRODUCTION TO MATHEMATICS number of periodic waves in a vibrating ether, and the steady sound to similar waves in a vibrating air. It is not our purpose here to explain the theory of light or the theory of sound. We have said enough to make it evident that one of the first steps necessary to make mathematics a fit instrument for the investigation of Nature is that it should be able to express the essential periodicity of things. If we have grasped this, we can understand the importance of the mathe- matical conceptions which we have next to consider, namely, periodic functions. CHAPTER XIII TRIGONOMETRY TRIGONOMETRY did not take its rise from the general consideration of the periodicity of nature. In this respect its history is analo- gous to that of conic sections, which also had their origin in very particular ideas. Indeed, a comparison of the histories of the two sciences yields some very instructive analogies and contrasts. Trigonometry, like conic sec- tions, had its origin among the Greeks. Its inventor was Hipparchus (born about 160 B.C.), a Greek astronomer, who made his observations at Rhodes. His services to astronomy were very great, and it left his hands a truly scientific subject with important results established, and the right method of progress indicated. Perhaps the invention of trigonometry was not the least of these services to the main science of his study. The next man who extended trigonometry was Ptolemy, the great Alexandrian astronomer, whom we have already mentioned. We now 173 174 INTRODUCTION TO MATHEMATICS see at once the great contrast between conic sections and trigonometry. The origin of trigonometry was practical ; it was invented because it was necessary for astronomical re- search. The origin of conic sections was purely theoretical. The only reason for its initial study was the abstract interest of the ideas involved. Characteristically enough conic sections were invented about 150 years earlier than trigonometry, during the very best period of Greek thought. But the im- portance of trigonometry, both to the theory and the application of mathematics, is only one of innumerable instances of the fruitful ideas which the general science has gained from its practical applications. We will try and make clear to ourselves what trigonometry is, and why it should be generated by the scientific study of astronomy. In the first place : What are the measure- ments which can be made by an astronomer ? They are measurements of time and measure- ments of angles. The astronomer may adjust a telescope (for it is easier to discuss the familiar instrument of modern astronomers) so that it can only turn about a fixed axis pointing east and west ; the result is that the telescope can only point to the south, with a greater or less elevation of direction, or, if turned round beyond the zenith, point to the north. This is the transit instrument, the TRIGONOMETRY 175 great instrument for the exact measurement of the times at which stars are due south or due north. But indirectly this instrument measures angles. For when the time elapsed between the transits of two stars has been noted, by the assumption of the uniform rotation of the earth, we obtain the angle through which the earth has turned in that period of time. Again, by other instruments, the angle between two stars can be directly measured. For if E is the eye of the astrono- Fig. 22. mer, and EA and EB are the directions in which the stars are seen, it is easy to devise instruments which shall measure the angle AEB. Hence, when the astronomer is form- ing a survey of the heavens, he is, in fact, measuring angles so as to fix the relative directions of the stars and planets at any in- stant. Again, in the analogous problem of 176 INTRODUCTION TO MATHEMATICS land-surveying, angles are the chief subject of measurements. The direct measurements of length are only rarely possible with any accuracy ; rivers, houses, forests, mountains, and general irregularities of ground all get in the way. The survey of a whole country will depend only on one or two direct measure- ments of length, made with the greatest elaboration in selected places like Salisbury Plain. The main work of a survey is the measurement of angles. For example, A, B, and C will be conspicuous points in the dis- Fig. 23. trict surveyed, say the tops of church towers. These points are visible each from the others. Then it is a very simple matter at A to measure the angle BAG, and at B to measure the angle ABC, and at C to measure the angle BCA. Theoretically, it is only necessary to measure two of these angles ; for, by a well- known proposition in geometry, the sum of the three angles of a triangle amounts to two TRIGONOMETRY 177 right-angles, so that when two of the angles are known, the third can be deduced. It is better, however, in practice to measure all three, and then any small errors of observa- tion can be checked. In the process of map- making a country is completely covered with triangles in this way. This process is called triangulation, and is the fundamental process in a survey. Now, when all the angles of a triangle are known, the shape of the triangle is known that is, the shape as distinguished from the size. We here come upon the great principle of geometrical similarity. The idea is very familiar to us in its practical applications. We are all familiar with the idea of a plan drawn to scale. Thus if the scale of a plan be an inch to a yard, a length of three inches in the plan means a length of three yards in the original. Also the shapes depicted in the plan are the shapes in the original, so that a right-angle in the original appears as a right- angle in the plan. Similarly in a map, which is only a plan of a country, the proportions of the lengths in the map are the proportions of the distances between the places indicated, and the directions in the map are the direc- tions in the country. For example, if in the map one place is north-north-west of the other, so it is in reality ; that is to say, in a map the angles are the same as in reality. 178 INTRODUCTION TO MATHEMATICS Geometrical similarity may be defined thus : Two figures are similar (i) if to any point in one figure a point in the other figure corresponds, so that to every line there is a corresponding line, and to every angle a corresponding angle, and (ii) if the lengths of corresponding lines are in a fixed propor- tion, and the magnitudes of corresponding angles are the same. The fixed proportion of the lengths of corresponding lines in a map (or plan) and in the original is called the scale of the map. The scale should always be indicated on the margin of every map and plan. It has already been pointed out that two triangles whose angles are respectively equal are similar. Thus, if the two triangles 6 E' C F Fig. 24. ABC and DEF have the angles at A and D equal, and those at B and E, and those at C and F, then DE is to AB in the same propor- TRIGONOMETRY 179 tion as EF is to BC, and as FD is to CA. But it is not true of other figures that simi- larity is guaranteed by the mere equality of angles. Take for example, the familiar cases of a rectangle and a square. Let ABCD be a square, and ABEF be a rectangle. Then all the corresponding angles are equal. But B Fig. 25. whereas the side AB of the square is equal to the side AB of the rectangle, the side EC of the square is about half the size of the side BE of the rectangle. Hence it is not true that the square ABCD is similar to the rect- angle ABEF. This peculiar property of the triangle, which is not shared by other recti- linear figures, makes it the fundamental figure in the theory of similarity. Hence in surveys, triangulation is the fundamental process ; and hence also arises the word " tri- 180 INTRODUCTION TO MATHEMATICS gonometry," derived from the two Greek words trigonon a triangle and metria measure- ment. The fundamental question from which trigonometry arose is this : Given the magni- tudes of the angles of a triangle, what can be stated as to the relative magnitudes of the sides. Note that we say " relative magnitudes of the sides," since by the theory of similarity it is only the proportions of the sides which are known. In order to answer this ques- tion, certain functions of the magnitudes of an angle, considered as the argument, are in- troduced. In their origin these functions were got at by considering a right-angled tri- angle, and the magnitude of the angle was defined by the length of the arc of a circle. In modern elementary books, the funda- mental position of the arc of the circle as de- fining the magnitude of the angle has been pushed somewhat to the background, not to the advantage either of theory or clearness of explanation. It must first be noticed that, in relation to similarity, the circle holds the same fundamental position among curvi- linear figures, as does the triangle among rectilinear figures. Any two circles are simi- lar figures ; they only differ in scale. The lengths of the circumferences of two circles, such as APA' and A-J*\A\ in the fig. 26 are in proportion to the lengths of their radii. Furthermore, if the two circles have the same TRIGONOMETRY 181 centre 0, as do the two circles in fig. 26, then the arcs AP and A\P\ intercepted by the arms of any angle AOP, are also in propor- tion to their radii. Hence the ratio of the Fig. 26. length of the arc AP to the length of the arc A.P radius OP, that is 7^= is a number which radius OP is quite independent of the length OP, and is the same as the fraction -^ ^4- . This f rac- radms OPi tion of " arc divided by radius " is the proper theoretical way to measure the magnitude of 182 INTRODUCTION TO MATHEMATICS an angle ; for it is dependent on no arbitrary unit of length, and on no arbitrary way of dividing up any arbitrarily assumed angle, AP such as a right-angle. Thus the fraction represents the magnitude of the angle AOP. Now draw PM perpendicularly to OA. Then the Greek mathematicians called the line PM the sine of the arc AP, and the line OM the cosine of the arc AP. They were well aware that the importance of the relations of these various lines to each other was dependent on the theory of similarity which we have just expounded. But they did not make their definitions express the properties which arise from this theory. Also they had not in their heads the modern general ideas respecting functions as correlating pairs of variable num- bers, nor in fact were they aware of any modern conception of algebra and algebraic analysis. Accordingly, it was natural to them to think merely of the relations between certain lines in a diagram. For us the case is different : we wish to embody our more powerful ideas. Hence, in modern mathematics, instead of considering the arc AP, we consider AP the fraction , which is a number the same for all lengths of OP ; and, instead of considering the lines PM and OM, we con- TRIGONOMETRY 183 PM , OM , . , sider the fractions and -, which again are numbers not dependent on the length of OP, i.e. not dependent on the scale of our PM diagrams. Then we define the number PA to be the sine of the number -, and the number =- to be the cosine of the number PA These fractional forms are clumsy to AP print ; so let us put u for the fraction which represents the magnitude of the angle PM AOP, and put v for the fraction -, and w OM for the fraction - Then u, v, w, are num- bers, and, since we are talking of any angle AOP, they are variable numbers. But al correlation exists between their magnitudes,! so that when u (i.e. the angle AOP) is given, the magnitudes of v and w are definitely deter-! mined. Hence v and w are functions of the argument u. We have called v the sine of u, and w the cosine of u. We wish to adapt the general functional notation y=f(x) to these special cases : so in modern mathe- matics we write sin for " / " when we want to 184 INTRODUCTION TO MATHEMATICS indicate the special function of " sine," and "cos" for "/" when we want to indicate the special function of "cosine." Thus, with the above meanings for u, v, w, we get v=sin u, and w=cos u, where the brackets surrounding the x in /(#) are omitted for the special functions. The meaning of these functions sin and cos as correlating the pairs of numbers u and v, and u and w is, that the functional relations are to be found by constructing (cf. fig. 26) an angle AOP, whose measure " AP divided by OP " is equal to u, and that then v is the number given by " PM divided by OP " and w is the number given by " OM divided by OP." It is evident that without some further defi- nitions we shall get into difficulties when the number u is taken too large. For then the arc AP may be greater than one-quarter of the circumference of the circle, and the point M (cf. figs. 26 and 27) may fall between and A' and not between O and A. Also P may be below the line AOA' and not above it as in fig. 26. In order to get over this difficulty we have recourse to the ideas and conven- tions of coordinate geometry in making our complete definitions of the sine and cosine. Let one arm OA of the angle be the axis OX, and produce the axis backwards to obtain its negative part OX'. Draw the TRIGONOMETRY 185 other axis YOY' perpendicular to it. Let any point P at a distance r from O have coordinates x and y. These coordinates are both positive in the first "quadrant" of the plan, e.g. the coordinates x and y of P in fig. 27. In the other quadrants, either one or both of the coordinates are negative, for example, x' and y for P', and x' and y' for P", and x and y' for P'" in fig. 27, where a?' and y' are both negative numbers. The positive angle POA is the arc AP divided by r, its sine is - and its cosine is - ; the posi- tive angle AOP' is the arc ABP' divided by r, its sine is - and cosine - ; the positive angle AOP" is the arc ABA'P" divided by r, its t/' x' sine is - and its cosine is - ; the positive r r angle AOP'" is the arc ABA'B'P'" divided w' ^E by r, its sine is - and its cosine is -. r r But even now we have not gone far enough. For suppose we choose t* to be a number greater than the ratio of the whole circum- ference of the circle to its radius. Owing to the similarity of all circles this ratio is the same for all circles. It is always denoted in mathematics by the symbol 2ir t where TT is the Greek form of the letter p and its name in the Greek alphabet is " pi." It can be proved that TT is an incommensurable number, and that therefore its value cannot be expressed by any fraction, or by any terminating or recurring decimal. Its value to a few decimal places is 3-14159 ; for many purposes a sufficiently accurate approximate value is . Mathematicians can easily cal- 7 culate TT to any degree of accuracy required, just as \/2 can be so calculated. Its value has been actually given to 707 places of TRIGONOMETRY 187 decimals. Such elaboration of calculation is merely a curiosity, and of no practical or theoretical interest. The accurate deter^ mination of TT is one of the two parts of the famous problem of squaring the circle. The other part of the problem is, by the theoretical methods of pure geometry to describe a straight line equal in length to the circumference. Both parts of the problem are now known to be impossible ; and the insoluble problem has now lost all special practical or theoretical interest, having be- ' come absorbed in wider ideas. After this digression on the value of TT, we now return to the question of the general definition of the magnitude of an angle, so as to be able to produce an angle corresponding to any value u. Suppose a moving point, Q, to start from A on OX (cf . fig. 27), and to rotate in the positive direction (anti-clockwise, in the figure considered) round the circumference of the circle for any number of times, finally resting at any point, e.g. at P or P' or P" or P"'. Then the total length of the curvilinear circular path traversed, divided by the radius of the circle, r, is the generalized definition of ( a positive angle of any size. Let x, y be the coordinates of the point in which the point Q rests, i.e.in one of the four alternative positions mentioned in fig. 27 ; x and y (as here used) will either x and y, or x r and y, or x' and t/', or x 188 INTRODUCTION TO MATHEMATICS and y'. Then the sign of this generalized a i flfl angle is - and its cosine is -. With these r r definitions the functional relations a=sin u and ry=cos u, are at last defined for all posi- tive real values of u. For negative values of u we simply take rotation of Q in the opposite (clockwise) direction ; but it is not worth our while to elaborate further on this point, now that the general method of procedure has been explained. These functions of sine and cosine, as thus defined, enable us to deal with the problems concerning the triangle from which Trigono- metry took its rise. But we are now in a position to relate Trigonometry to the wider idea of Periodicity of which the importance was explained in the last chapter. It is easy to see that the functions sin u and cos u are periodic functions of u. For consider the position, P (in fig. 27), of a moving point, Q, which has started from A and revolved round the circle. This position, P, marks the angles arc AP , , . arc AP , A . arc AP - , and 2 TT-\ --- , and 4 ir-\ -- , Q Trt and 6 TT + , and so on indefinitely. Now, all these angles have the same sine and cosine, ?/ 72 namely, and -. Hence it is easy to see that, TRIGONOMETRY 189 if u be chosen to have any value, the argu- ments u and 2 TT+U, and 4-77+ w, and 67r-fw, and STT+U and so on indefinitely, have all the same values for the corresponding sines and cosines. In other words, sn w=sn 2 =etc. ; COS U = COS (27T+tt)=COS (47T+w)=COS =etc. This fact is expressed by saying that s in u and cos u are periodic functions with their period equal to 2?r. The graph of the function t/=sin x (notice that we now abandon v and u for the more familiar y and x) is shown in fig. 28. We take on the axis of x any arbitrary length at pleasure to represent the number TT, and on the axis of y any arbitrary length at pleasure to -repre- sent the number 1. The numerical values of the sine and cosine can never exceed unity. The recurrence of the figure after periods of 2-7T will be noticed. This graph represents the simplest style of periodic function, out of which all others are constructed. The cosine gives nothing fundamentally different from the sine. For it is easy to prove that cos x= rjf sin (# + -) ; hence it can be seen that the it graph of cos x is simply fig. 28 modified by 190 INTRODUCTION TO MATHEMATICS drawing the axis of OF through the point on OX marked -, instead of drawing it in its actual position on the figure. It is easy to construct a ' sine ' function in Fig. 28. which the period has any assigned value a. For we have only to write . 27TX and then sin (27TX . _ l -- \-2ir [a I . STHC 1= sin . d Thus the period of this new function is now a. Let us now give a general definition of what TRIGONOMETRY 191 we mean by a periodic function. The function /(a?) is periodic, with the period a, if (i) for any value of x we have f(x)=f(x-{-a), and (ii) there is no number b smaller than a such that for any value of x, f(x)=f(x-}-b). The second clause is put into the definition because when we have sin - , it is not only a periodic in the period a, but also in the periods 2a and 3a, and so on ; this arises since . 27r(#+3a) . /2-rrx , \ . 2irx sin - ! - ' = sml -- h67r 1 =sm - . a \ a / a So it is the smallest period which we want to get hold of and call the period of the function. The greater part of the abstract theory of periodic functions and the whole of the appli- cations of the theory to Physical Science are dominated by an important theorem called Fourier's Theorem ; namely that, if /(#) be a periodic function with the period a and if f(x) also satisfies certain conditions, which practic- ally are always presupposed in functions sug- gested by natural phenomena, then /(a?) can be written as the sum of a set of terms in the form 2TTX . \ , . /47T -f j j +c 2 sin \r . iG-rrx , \ in I -- 1-^3) . , , +c 3 sin I -- 1-^3)+ etc. 192 INTRODUCTION TO MATHEMATICS In this formula CQ, GI, c%, 03, etc., and also fit %> 3* etc., are constants, chosen so as to suit the particular function. Again we have to ask, How many terms have to be chosen ? And here a new difficulty arises : for we can prove that, though in some particular cases a definite number will do, yet in general all we can do is to approximate as closely as we like to the value of the function by taking more and more terms. This process of gradual approximation brings us to the consideration of the theory of infinite series, an essential part of mathematical theory which we will consider in the next chapter. The above method of expressing a periodic function as a sum of sines is called the " har- monic analysis " of the function. For ex- ample, at any point on the sea coast the tides rise and fall periodically. Thus at a point near the Straits of Dover there will be two daily tides due to the rotation of the earth. The daily rise and fall of the tides are com- plicated by the fact that there are two tidal waves, one coming up the English Channel, and the other which has swept round the North of Scotland, and has then come south- ward down the North Sea. Again some high tides are higher than others : this is due to the fact that the Sun has also a tide-generating influence as well as the Moon. In this way monthly and other periods are introduced. TRIGONOMETRY 193 We leave out of account the exceptional in- fluence of winds which cannot be foreseen. The general problem of the harmonic analysis of the tides is to find sets of terms like those in the expression on page 191 above, such that each set will give with approximate accuracy the contribution of the tide-generating influ- ences of one " period " to the height of the tide at any instant. The argument x will therefore be the time reckoned from any con- venient commencement. Again, the motion of vibration of a violin string is submitted to a similar harmonic analysis, and so are the vibrations of the ether and the air, corresponding respectively to waves of light and waves of sound. We are here in the presence of one of the funda- mental processes of mathematical physics namely, nothing less than its general method of dealing with the great natural fact of Periodicity. CHAPTER XIV SERIES No part of Mathematics suffers more from the triviality of its initial presentation to beginners than the great subject of series. Two minor examples of series, namely arith- metic and geometric series, are considered ; these examples are important because they are the simplest examples of an important general theory. But the general ideas are never disclosed ; and thus the examples, which exemplify nothing, are reduced to silly triviali- ties. The general mathematical idea of a series is that of a set of things ranged in order, that is, in sequence; This meaning is accurately represented in the common use of the term. Consider for example, the series of English Prime Ministers during the nineteenth century, arranged in the order of their first tenure of that office within the century. The series commences with William Pitt, and ends with Lord Rosebery, who, appropriately enough, is the biographer of the first member. We 194 SERIES 195 might have considered other serial orders for the arrangement of these men ; for example, according to their height or their weight. These other suggested orders strike us as trivial in connection with Prime Ministers, and would not naturally occur to the mind ; but abstractly they are just as good orders as any other. When one order among terms is very much more important or more obvious than other orders, it is often spoken of as the order of those terms. Thus the order of 'the integers would always be taken to mean their order as arranged in order of magnitude. But of course there is an indefinite number -of other ways of arranging them. When the number of things considered is finite, the number of ways of arranging them in order is called the number of their permutations. The number of permutations of a set of n things, where n is some finite integer, is nx(n l)x(n 2)x(n 3)x...x4x3x2xl that is to say, it is the product of the first n integers ; this product is so important in mathematics that a special symbolism, is used for it, and it is always written ' n 1 ' Thus, 21=2x1=2, and 3!=3x2xl=6, and 4!=4 x3x2xl=24, and 51=5x4x3x2x1=120. As n increases, the value of n \ increases very quickly ; thus 100 ! is a hundred times as large as 99 I 196 INTRODUCTION TO MATHEMATICS It is easy to verify in the case of small values of n that n ! is the number of ways of arranging n things in order. Thus con- sider two things a and b ; these are capable of the two orders ab and ba, and 2 ! =2. Again, take three things a, b, and c ; these are capable of the six orders, abc, acb, bac, bca, cab, cba t and 31=6. Similarly for the twenty-four orders in which four things a, b, c, and d, can be arranged. When we come to the infinite sets of things like the sets of all the integers, or all the fractions, or all the real numbers for instance we come at once upon the complications of the theory of order-types. This subject was touched upon in Chapter VI. in considering the possible orders of the integers, and of the fractions, and of the real numbers. The whole question of order-types forms a com- paratively new branch of mathematics of great importance. We shall not consider it any further. All the infinite series which we consider now are of the same order-type as the integers arranged in ascending order of magnitude, namely, with a first term, and such that each term has a couple of next- door neighbours, one on either side, with the exception of the first term which has, of course, only one next-door neighbour. Thus, if m be any integer (not zero), there will be always an mth term. A series with a finite SERIES 197 number of terms (say n terms) has the same characteristics as far as next-door neighbours are concerned as an infinite series ; it only differs from infinite series in having a last term, namely, the nth. The important thing to do with a series of numbers using for the future " series " in the restricted sense which has just been men- tioned is to add its successive terms to- gether. Thus if u\, Uz, 3, . . . u n . . . are respec- tively the 1st, 2nd, 3rd, 4th, . . . nth, . . . terms of a series of numbers, we form succes- sively the series u\ t u\+uz, ^1+^2+^3, i-f- W2+W3+W4, and so on ; thus the sum of the 1st n terms may be written. If the series has only a finite number of terms, we come at last in this way to the sum of the whole series of terms. But, if the series has an infinite number of terms, this process of successively forming the sums of the terms never terminates ; and in this sense there is no such thing as the sum of an infinite series. But why is it important successively to add the terms of a series in this way ? The answer is that we are here symbolizing the funda- mental mental process of approximation. This is a process which has significance far 198 INTRODUCTION TO MATHEMATICS beyond the regions of mathematics. Our limited intellects cannot deal with compli- cated material all at once, and our method of arrangement is that of approximation. The statesman in framing his speech puts the dominating issues first and lets the details fall naturally into their subordinate places. There is, of course, the converse artistic method of preparing the imagination by the presentation of subordinate or special details, and then gradually rising to a crisis. In either way the process is one of gradual sum- mation of effects ; and this is exactly what is done by the successive summation of the terms of a series. Our ordinary method of stating numbers is such a process of gradual summation, at least, in the case of large numbers. Thus 568,213 presents itself to the mind as 500,000 +60,000 +8,000 +200 +10 +3 In the case of decimal fractions this is so more avowedly. Thus 3-14159 is Also, 3 and 3+^, and 3+^+^, and ~t~Ttfff ~^T?nn7 anc * ^+T 1 o'i'T^4~T7nn7~l~TTmn7 are successive approximations to the complete re- sult 3-14159. If we read 568,218 backwards from right to left, starting with the 3 units, SERIES 199 we read it in the artistic way, gradually pre- paring the mind for the crisis of 500,000. The ordinary process of numerical multi- plication proceeds by means of the summa- tion of a series, Consider the computation 342 658 2736 1710 2052 225036 Hence the three lines to be added form a series of which the first term is the upper line. This series follows the artistic method of presenting the most important term last, not from any feeling for art, but because of the convenience gained by keeping a firm hold on the units' place, thus enabling us to omit some O's, formally necessary. But when we approximate by gradually adding the successive terms of an infinite series, what are we approximating to ? The difficulty is that the series has no " sum " in the straightforward sense of the word, because the operation of adding together its terms can never be completed. The answer is that we are approximating to the limit of the summation of the series, and we must now 200 INTRODUCTION TO MATHEMATICS proceed to explain what the " limit " of a series is. The summation of a series approximates to a limit when the sum of any number of its terms, provided the number be large enough, is as nearly equal to the limit as you care to approach. But this description of the mean- ing of approximating to a limit evidently will not stand the vigorous scrutiny of modern mathematics. What is meant by large enough, and by nearly equal, and by care to approach ? All these vague phrases must be explained in terms of the simple abstract ideas which alone are admitted into pure mathematics. Let the successive terms of the series be i, U2, Ws, W4, . . . , u n) etc., so that u n is the nth term of the series. Also let s n be the sum of the 1st n terms, whatever n may be. So that and Then the terms $1, $2, $3, . . . $ n , . . . form a new series, and the formation of this series is the process of summation of the original series. Then the " approximation " of the summation of the original series to a " limit " means the " approximation of the terms of this new series to a limit." And we have SERIES 201 now to explain what we mean by the approxi- mation to a limit of the terms of a series. Now, remembering the definition (given in chapter XII.) of a standard of approxima- tion, the idea of a limit means this : I is the limit of the terms of the series si, $2, *s s n , . . ., if, corresponding to each real number k, taken as a standard of approximation, a term s n of the series can be found so that all succeeding terms (i.e. s n+i> *n+2> e * c -) approximate to I within that standard of approximation. If another smaller standard k 1 be chosen, the term s n may be too early in the series, and a later term 8 m with the above property will then be found. If this property holds, it is evident that as you go along to series Si, $2, $3, . . ., s n , . . . from left to right, after a time you come to terms all of which are nearer to I than any number which you may like to assign. In other words you approximate to I as closely as you like. The close connection of this definition of the limit of a series with the definition of a continuous function given in chapter XI. will be immediately perceived. Then coming back to the original series MI, t*2 3, . . . u n , . . ., the limit of the terms of the series Si, $2, $3, . . , 9 s n , . . ., is called the " sum to infinity " of the original series. But it is evident that this use of the word 202 INTRODUCTION TO MATHEMATICS " sum " is very artificial, and we must not assume the analogous properties to those of the ordinary sum of a finite number of terms without some special investigation. Let us look at an example of a " sum to infinity." Consider the recurring decimal 1111. . . . This decimal is merely a way of symbolizing the "sum to infinity " of the series 1, -01, -001, -0001, etc. The correspond- ing series found by summation is si = -I t $2 ='11, 53 ='111, 54 =-1111, etc. The limit of the terms of this series is ; this is easy to see by simple division, for ^=a+ 7 V=-ll+^=.lll+ irzr Vir= etc. Hence, if T 3 T is given (the k of the definition), 1 and all succeeding terms differ from by less than T 8 T ; if -^^ is given (another choice for the k of the definition), -111 and all succeeding terms differ from by less than YflVo-; and so on, whatever choice for k be made. It is evident that nothing that has been said gives the slightest idea as to how the "sum to infinity" of a series is to be found. We have merely stated the condi- tions which such a number is to satisfy. In- deed, a general method for finding in all cases the sum to infinity of a series is intrinsic- ally out of the question, for the simple reason that such a " sum," as here defined, does not always exist. Series which possess a sum to SERIES 203 infinity are called convergent, and those which do not possess a sum to infinity are called divergent. An obvious example of a divergent series is 1, 2, 3, . . ., n . . . i.e. the series of in- tegers in their order of magnitude. For whatever number I you try to take as its sum to infinity, and whatever standard ol approximation A; you choose, by taking enough terms of the series you can always make their sum differ from / by more than k. Again, another example of a divergent series is 1, 1, 1, etc., i.e. the series ol which each term is equal to 1. Then the sum of n terms is n, and this sum grows without limit as n increases. Again, another example of a divergent series is 1, 1, 1, 1, 1, 1, etc., i.e. the series in which the terms are alternately 1 and 1. The sum of an odd number of terms is 1, and of an even number of terms is 0. Hence the terms of the series $1, $2, $3, . . . s n , . . . do not ap- proximate to a limit, although they do not increase without limit. It is tempting to suppose that the condi- tion for MI, 2 u nt . . . to have a sum to infinity is that u n should decrease inde- finitely as n increases. Mathematics would be a much easier science than it is, if this were the case. Unfortunately the supposition is not true. 204 INTRODUCTION TO MATHEMATICS For example the series 111 1 7 2' 3' 4' ' ' *' n ' * ' is divergent. It is easy to see that this is the case ; for consider the sum of n terms ginning at the (n+1)" 1 term. These n 4-U terms are - -, - -, - -, ...--: there w+l'n+2'n+3' 2n are n of them and is the least among them. SBO Hence their sum is greater than n times , i.e. is greater than -. Now, without altering the sum to infinity, if it exist, we can add together neighbouring terms, and obtain the series that is, by what has been said above, a series whose terms after the 2nd are greater than those of the series, 1, i, i, I, etc., where all the terms after the first are equal. But this series is divergent. Hence the original series is divergent.* This question of divergency shows how careful we must be in arguing from the pro- * Cf. Note C, p. 251. SERIES 205 perties of the sum of a finite number of terms to that of the sum of an infinite series. For the most elementary property of a finite number of terms is that of course they possess a sum : but even this fundamental property is not necessarily possessed by an infinite series. This caution merely states that we must not be misled by the suggestion of the technical term " sum of an infinite series." It is usual to indicate the sum of the infinite series t*i, u 2 , t*s, . . . t* n . . . . by We now pass on to a generalization of the idea of a series, which mathematics, true to its method, makes by use of the variable. Hitherto, we have only contemplated series in which each definite term was a definite number. But equally well we can generalize, and make each term to be some mathematical expression containing a variable x. Thus we may consider the series 1, x, x 2 , x* t . . ., x n t . . ., and the series x 2 - x 3 x" *' ' 8" ..... 7P ' ' ' In order to symbolize the general idea of any such function, conceive of a function of x, f n (x) say, which involves in its formation a variable integer n, then, by giving n the 206 INTRODUCTION TO MATHEMATICS values 1, 2, 3, etc., in succession, we get the series /i(), h(x), h(x), . . ., f n (x), . . . Such a series may be convergent for some values of as and divergent for others. It is, in fact, rather rare to find a series involving a variable x which is convergent for all values of x, at least in any particular instance it is very unsafe to assume that this is the case. For example, let us examine the simplest of all instances, namely, the " geometrical " series 1, x, x 2 , # 3 , . . ., x n , . . . The sum of n terms is given by Now multiply both sides by x and we get Now subtract the last line from the upper line and we get s n (l x) =s n xs n =1 aP+\ and hence (if x be not equal to 1) 1 gn+l . 1 x n+l ^ __ n I-x ~ l^x ~T^x Now if x be numerically less than 1, for suffi- jn+l ciently large values of n, - - is always numeri- L X J SERIES 20T cally less than k, however k be chosen. Thus, if x be numerically less than 1, the series 1, #, # 2 , . . . tc n , . . . is convergent, and - - is its 1 x limit. This statement is symbolized by . . ., (-1 <x A X But if x' is numerically greater than 1, or numerically equal to 1, the series is divergent. In other words, if x lie between 1 and -f-l> the series is convergent ; but if x be equal to 1 or -}-l, or if a; lie outside the interval 1 to +1, then the series is divergent. Thus the series is convergent at all " points " within the interval 1 to +i> exclusive of the end points. At this stage of our enquiry another ques- tion arises. Suppose that the series is convergent for all values of x lying within the interval a to b, i.e. the series is convergent for any value of x which is greater than a and less than b. Also, suppose we want to be sure that in approximating to the limit we add together enough terms to come within some standard of approximation k. Can we always state some number of terms, say n, such that, if we take n or more terms to form the sum, then whatever value x has 208 INTRODUCTION TO MATHEMATICS within the interval we have satisfied the desired standard of approximation? Sometimes we can and sometimes we can- not do this for each value of k. When we can, the series is called uniformly convergent throughout the interval, and when we cannot do so, the series is called non-uniformly con- vergent throughout the interval. It makes a great difference to the properties of a series whether it is or is not uniformly convergent through an interval. Let us illustrate the matter by the simplest example and the simplest numbers. Consider the geometric series It is convergent throughout the interval 1 to +1, excluding the end values aj= 1. But it is not uniformly convergent through- out this interval. For if s n (x) be the sum of n terms, we have proved that the difference 1 # n+1 between s n (x) and the limit - - is - - 1 a? \x Now suppose n be any given number of terms, say 20, and let k be any assigned standard of approximation, say -001. Then, by taking $ near enough to + 1 or near enough to 1, #21 we can make the numerical value of - - to Ia? be greater than -001. Thus 20 terms will SERIES 209 not do iver the whole interval, though it is more thai enough over some parts of it. The sane reasoning can be applied what- ever other number we take instead of 20, and whatever standard of approximation in- stead of -001. Hence the geometric series I-\-x+x 2 -{-x 3 -t . . . +x tt + ... is non-uni- formly convergent over its whole interval of convergence 1 to -f-1. But if we take any smaller interval lying at both ends within the interval 1 to +1, the geometric series is uniformly convergent within it. For ex- ample, take the interval to +^. Then any value for n which makes - - numerically 1 a; less than k at these limits for x also serves for all values of x between these limits, since it so happens that diminishes in numeri- 1 x cal value as x diminishes in numerical value. For example, take k =-001; then, putting x = vzr . we find : forn=l, 1 X 1 Tr -rn+1 / 1 \3 for n=2, - = ^r =ir^= -00111 . . ., 1 X 1 rfr f or n= 3, 5 = J^ = W V7= '000111 . . ., 1 x 1 yir Thus three terms will do for the whole in- 210 INTRODUCTION TO MATHEMATICS terval, though, of course, for some parts of the interval it is more than is necessary. Notice that, because l+a4-# 2 -}- . . . + n + ... is convergent (though not uni- formly) throughout the interva 1 1 to +1, for each value of x in the internal some num- ber of terms n can be found wMch will satisfy a desired standard of approximation ; but, as we take x nearer and nearer to either end value +1 or 1, larger and larger values of n have to be employed. It is curious that this important distinction between uniform and non-uniform conver- gence was not published till 1847 by Stokes afterwards, Sir George Stokes and later, in- dependently in 1850 by Seidel, a German mathematician. The critical points, where non-uniform con- vergence comes in, are not necessarily at the limits of the interval throughout which con- vergence holds. This is a speciality belonging to the geometric series. In the case of the geometric series l+# . . +x n + . . ., a simple algebraic expression - - can be given for its limit in 1 x its interval of convergence. But this is not always the case. Often we can prove a series to be convergent within a certain interval, though we know nothing more about its limit except that it is the limit of the series. SERIES 211 But this is a very good way of defining a function ; viz. as the limit of an infinite con- vergent series, and is, in fact, the way in which most functions are, or ought to be, defined. Thus, the most important series in ele- mentary analysis is where n \ has the meaning defined earlier in this chapter. This series can be proved to be absolutely convergent for all values of , and to be uniformly convergent within any interval which we like to take. Hence it has all the comfortable mathematical properties which a series should have. It is called the exponential series. Denote its sum to infinity by exp#. Thus, by definition, expa? is called the exponential function. It is fairly easy to prove, with a little knowledge of elementary mathematics, that (expa?)x(expt/)=exp(#+t/) . . .(A) In other words that 212 INTRODUCTION TO MATHEMATICS This property (A) is an example of what is called an addition-theorem. IVhen any function [say /(#)] has been denned, the first thing we do is to try to express /(#+*/) in terms of known functions of x only, and known func- tions of y only. If we can do so, the result is called an addition-theorem. Addition- theorems play a great part in mathematical analysis. Thus the addition-theorem for the sine is given by sin (x+y)sin x cos y+cos x sin y, and for the cosine by cos (x+y) = cos x cos y sin x sin y. As a matter of fact the best ways of de- fining sin x and cos x are not by the elaborate geometrical methods of the previous chapter, but as the limits respectively of the series x 3 x 5 x so that we put x 3 . x 5 x 7 . sin *=*--+___ +etc , X 2 , X 4 X Q , cos *=!-_+- +etc SERIES 218 These definitions are equivalent to the geo- metrical definitions, and both series can be proved to be convergent for all values of a?, and uniformly convergent throughout any interval. These series for sine and cosine have a general likeness to the exponential series given above. They are, indeed, intim- ately connected with it by means of the theory of imaginary numbers explained in Chapters VII. and VIII. X. A * Fig. 29. The graph of the exponential function is given in fig. 29. It cuts the axis OF at the point t/=l, as evidently it ought to do, since when x=Q every term of the series except the first is zero. The importance of the ex- ponential function is that it represents any changing physical quantity whose rate of increase at any instant is a uniform per- centage of its value at that instant. For 214 INTRODUCTION TO MATHEMATICS example, the above graph represents the size at any time of a population with a uniform birth-rate, a uniform death-rate, and no emi- gration, where the x corresponds to the time reckoned from any convenient day, and the y represents the population to the proper scale. The scale must be such that OA re- presents the population at the date which is taken as the origin. But we have here come upon the idea of " rates of increase " which is the topic for the next chapter. An important function nearly allied to the exponential function is found by putting aj 2 for x as the argument in the exponential func- tion. We thus get exp. ( x 2 ). The graph t/=exp. ( x 2 ) is given in fig. 30. Fig. 30. The curve, which is something like a cocked hat, is called the curve of normal error. Its SERIES 215 corresponding function is vitally important to the theory of statistics, and tells us in many cases the sort of deviations from the average results which we are to expect. Another important function is found by combining the exponential function with the sine, in this way : y =exp( car) xsin 2772! Fig. 31. Its graph is given in fig. 31. The points A, B, O, C, D, E, F, are placed at equal in- tervals \p, and an unending series of them should be drawn forwards and backwards. This function represents the dying away of vibrations under the influence of friction or of " damping " forces. Apart from the friction, the vibrations would be periodic, with a period p ; but the influence of the friction 216 INTRODUCTION TO MATHEMATICS makes the extent of each vibration smaller than that of the preceding by a constant per- centage of that extent. This combination of the idea of " periodicity " (which requires the sine or cosine for its symbolism) and of " constant percentage " (which requires the exponential function for its symbolism) is the reason for the form of this function, namely, its form as a product of a sine-function into an exponential function. CHAPTER XV THE DIFFERENTIAL CALCULUS THE invention of the differential calculus marks a crisis in the history of mathematics, The progress of science is divided between periods characterized by a slow accumulation of ideas and periods, when, owing to the new material for thought thus patiently collected, some genius by the invention of a new method or a new point of view, suddenly transforms the whole subject on to a higher level. These contrasted periods in the progress of the history of thought are compared by Shelley to the formation of an avalanche. The sun-awakened avalanche ! whose mass, Thrioe sifted by the storm, had gathered there Flake after flake, in heaven-defying minds As thought by thought is piled, till some great truth Is loosened, and the nations echo round, The comparison will bear some pressing. The final burst of sunshine which awakens the avalanche is not necessarily beyond com- parison in magnitude with the other powers of nature which have presided over its slow 217 218 INTRODUCTION TO MATHEMATICS formation. The same is true in science. The genius who has the good fortune to produce the final idea which transforms a whole region of thought, does not necessarily excel all his predecessors who have worked at the preliminary formation of ideas. In consider- ing the history of science, it is both silly and ungrateful to confine our admiration with a gaping wonder to those men who have made the final advances towards a new epoch In the particular instance before us, the subject had a long history before it as- sumed its final form at the hands of its two inventors. There are some traces of its methods even among the Greek mathe- maticians, and finally, just before the actual production of the subject, Fermat (born 1601 A.D., and died 1665 A.D.), a distinguished French mathematician, had so improved on previous ideas that the subject was all but created by him. Fermat, also, may lay claim to be the joint inventor of coordinate geometry in company with his contemporary and countryman, Descartes. It was, in fact, Descartes from whom the world of science received the new ideas, but Fermat had cer- tainly arrived at them independently. We need not, however, stint our admira- tion either for Newton or for Leibniz. New- ton was a mathematician and a student of physical science, Leibniz was a mathema- DIFFERENTIAL CALCULUS 219 tician and a philosopher, and each of them in his own department of thought was one of the greatest men of genius that the world has known. The joint invention was the occasion of an unfortunate and not very creditable dispute. Newton was using the methods of Fluxions, as he called the subject, in 1666, and employed it in the composition of his Principia, although in the work as printed any special algebraic notation is avoided. But he did not print a direct state- ment of his method till 1693. Leibniz pub- lished his first statement in 1684. He was accused by Newton's friends of having got it from a MS. by Newton, which he had been shown privately. Leibniz also accused New- ton of having plagiarized from him. There is now not very much doubt but that both should have the credit of being independent discoverers. The subject had arrived at a stage in which it was ripe for discovery, and there is nothing surprising in the fact that two such able men should have independ- ently hit upon it. These joint discoveries are quite common in science. Discoveries are not in general made before they have been led up to by the previous trend of thought, and by that time many minds are in hot pursuit of the important idea. If we merely keep to discoveries in which Englishmen are 220 INTRODUCTION TO MATHEMATICS concerned, the simultaneous enunciation of the law of natural selection by Darwin and Wallace, and the simultaneous discovery of Neptune by Adams and the French astrono- mer, Leverrier, at once occur to the mind. The disputes, as to whom the credit ought to be given, are often influenced by an unworthy spirit of nationalism. The really inspiring reflection suggested by the history of mathe- matics is the unity of thought and interest among men of so many epochs, so many nations, and so many races. Indians, Egyptians, Assyrians, Greeks, Arabs, Italians, French- men, Germans, Englishmen, and Russians, have all made essential contributions to the pro- gress of the science. Assuredly the jealous exaltation of the contribution of one particu- lar nation is not to show the larger spirit. The importance of the differential calculus arises from the very nature of the subject, which is the systematic consideration of the rates of increase of functions. This idea is immediately presented to us by the study of nature ; velocity is the rate of increase of the distance travelled, and acceleration is the rate of increase of velocity. Thus the funda- mental idea of change, which is at the basis of our whole perception of phenomena, immedi- ately suggests the enquiry as to the rate of change. The familiar terms of " quickly " and " slowly " gain their meaning from a tacit DIFFERENTIAL CALCULUS 221 reference to rates of change. Thus the differ- ential calculus is concerned with the very key of the position from which mathematics can be successfully applied to the explanation of the course of nature. This idea of the rate of change was certainly in Newton's mind, and was embodied in the o T ft Fig. 32. language in which he explained the subject. It may be doubted, however, whether this point of view, derived from natural phenomena, was ever much in the minds of the preced- ing mathematicians who prepared the subject for its birth. They were concerned with the more abstract problems of drawing tangents to curves, of finding the lengths of curves, and of finding the areas enclosed by curves. The 222 INTRODUCTION TO MATHEMATICS last two problems, of the rectification of curves and the quadrature of curves as they are named, belong to the Integral Calculus, which is however involved in the same general subject as the Differential Calculus. The introduction of coordinate geometry makes the two points of view coalesce. For (cf. fig. 32) let AQP be any curved line and let PT be the tangent at the point P on it. Let the axes of coordinates be OX and OY ; and let y =/(#) be the equation to the curve, so that OM=x, and PM=y. Now let Q be any moving point on the curve, with coordinates #i */i> ; then yi =f(xi). And let Q' be the point on the tangent with the same abscissa x\ ; suppose that the coordinates of Q' are x\ and y'. Now suppose that N moves along the axis OX from left to right with a uniform velocity ; then it is easy to see that the ordi- nate y' of the point Q' on the tangent TP also increases uniformly as Q' moves along the tangent in a corresponding way. In fact it is easy to see that the ratio of the rate of increase of Q'N to the rate of increase of ON is in the ratio of Q'N to TN, which is the same at all points of the straight line. But the rate of increase of Q2V, which is the rate of increase of /(#i), varies from point to point of the curve so long as it is not straight. As Q passes through the point P, the rate of increase of / (#1) (where x\ coincides with x for the moment) is the same as the rate of increase of y' on the tangent at P. Hence, if we have a general method of determining the rate of increase of a function /(#) of a variable x, we can determine the slope of the tangent at any point (x, y,) on a curve, and thence can draw it. Thus the problems of drawing tan- gents to a curve, and of determining the rates of increase of a function are really identical. It will be noticed that, as in the cases of Conic Sections and Trigonometry, the more artificial of the two points of view is the one in which the subject took its rise. The really fundamental aspect of the science only rose into prominence comparatively late in the day. It is a well-founded historical genera- lization, that the last thing to be discovered in any science is what the science is really about. Men go on groping for centuries, guided merely by a dim instinct and a puzzled curiosity, till at last " some great truth is loosened." Let us take some special cases in order to familiarize ourselves with the sort of ideas which we want to make precise. A train is in motion how shall we determine its velocity at some instant, let us say, at noon ? We can take an interval of five minutes which includes noon, and measure how far the train has gone in that period. Suppose we find it to be five 224 INTRODUCTION TO MATHEMATICS miles, we may then conclude that the train was running at the rate of 60 miles per hour. But five miles is a long distance, and we cannot be sure that just at noon the train was moving at this pace. At noon it may have been running 70 miles per hour, and afterwards the break may have been put on. It will be safer to work with a smaller interval, say one minute, which includes noon, and to measure the space traversed during that period. But for some purposes greater accuracy may be required, and one minute may be too long. In practice, the necessary inaccuracy of our measurements makes it useless to take too small a period for measure- ment. But in theory the smaller the period the better, and we are tempted to say that for ideal accuracy an infinitely small period is required. The older mathematicians, in particular Leibniz, were not only tempted, but yielded to the temptation, and did say it. Even now it is a useful fashion of speech, provided that we know how to interpret it into the language of common sense. It is curious that, in his exposition of the founda- tions of the calculus, Newton, the natural scientist, is much more philosophical than Leibniz, the philosopher, and on the other hand, Leibniz provided the admirable nota- tion which has been so essential for the pro- gress of the subject. Now take another example within the region of pure mathematics. Let us proceed to find the rate of increase of the function x 2 for any value x of its argument. We have not yet really denned what we mean by rate of increase. We will try and grasp its meaning in relation to this particular case. When x increases to x +h, the function x 2 increases to (x-\-h) 2 ; so that the total increase has been (x-\-h) 2 x 2 , due to an increase h in the argu- ment. Hence throughout the interval x to (x+ h) the average increase of the function per , . (x+h) 2 -x 2 unit increase of the argument is - ^~ - . But (x+h) 2 =x 2 +2hx+h* t and therefore nm ,, Thus 2x+h is the average increase of the function x 2 per unit increase in the argument, the average being taken over by the interval x to x+h. But 2x+h depends on h, the size of the interval. We shall evidently get what we want, namely the rate of increase at the value x of the argument, by diminishing h more and more. Hence in the limit when h 226 INTRODUCTION TO MATHEMATICS has decreased indefinitely, we say that 2x is the rate of increase of x 2 at the value x of the argument. Here again we are apparently driven up against the idea of infinitely small quantities in the use of the words " in the limit when h has decreased indefinitely." Leibniz held that, mysterious as it may sound, there were actu- ally existing such things as infinitely small quantities, and of course infinitely small num- bers corresponding to them. Newton's lan- guage and ideas were more on the modern lines ; but he did not succeed in explaining the matter with such explicitness so as to be evidently doing more than explain Leibniz's ideas in rather indirect language. The real explanation of the subject was first given by Weierstrass and the Berlin School of mathe- maticians about the middle of the nineteenth century. But between Leibniz and Weier- strass a copious literature, both mathematical and philosophical, had grown up round these mysterious infinitely small quantities which mathematics had discovered and philosophy proceeded to explain. Some philosophers, Bishop Berkeley, for instance, correctly denied the validity of the whole idea, though for reasons other than those indicated here. But the curious fact remained that, despite all criticisms of the foundations of the subject, there could be no doubt but that the mathe- DIFFERENTIAL CALCULUS 227 matical procedure was substantially right. In fact, the subject was right, though the explana- tions were wrong. It is this possibility of being right, albeit with entirely wrong ex- planations as to what is being done, that so often makes external criticism that is so far as it is meant to stop the pursuit of a method singularly barren and futile in the progress of science. The instinct of trained observers, and their sense of curiosity, due to the fact that they are obviously getting at something, are far safer guides. Anyhow the general effect of the success of the Differential Calculus was to generate a large amount of bad philo- sophy, centring round the idea of the in- finitely small. The relics of this verbiage may still be found in the explanations of many elementary mathematical text-books on the Differential Calculus. It is a safe rule to apply that, when a mathematical or philoso- phical author writes with a misty profundity, he is talking nonsense. Newton would have phrased the question by saying that, as h approaches zero, in the limit 2x+h becomes 2#. It is our task so to explain this statement as to show that it does not in reality covertly assume the existence of Leibniz's infinitely small quantities. In reading over the Newtonian method of state- ment, it is tempting to seek simplicity by 228 INTRODUCTION TO MATHEMATICS saying that 2x+h is 2x, when h is zero. But this will not do ; for it thereby abolishes the interval from as to x+ h, over which the average increase was calculated. The problem is, how to keep an interval of length h over which to calculate the average increase, and at the same time to treat h as if it were zero. Newton did this by the conception of a limit, and we now proceed to give Weiers trass's explanation of its real meaning. In the first place notice that, in discussing 2x +h, we have been considering x as fixed in value and h as varying. In other words x has been treated as a " constant " variable, or parameter, as explained in Chapter IX. ; and we have really been considering 2x+h as a function of the argument h. Hence we can generalize the question on hand, and ask what we mean by saying that the function /(/&) tends to the limit I, say, as its argument h tends to the value zero. But again we shall see that the special value zero for the argument does not belong to the essence of the subject ; and again we generalize still further, and ask, what we mean by saying that the function f(h) tends to the limit I as h tends to the value a. Now, according to the Weierstrassian ex- planation the whole idea of h tending to the value , though it gives a sort of metaphorical picture of what we are driving at, is really off the point entirely. Indeed it is fairly obvious that, as long as we retain anything like "A tending to a," as a fundamental idea, we are really in the clutches of the infinitely small ; for we imply the notion of h being infinitely near to a. This is just what we want to get rid of. Accordingly, we shall yet again restate our phrase to be explained, and ask what we mean by saying that the limit of the function f(h) at a is I. The limit of /(/) at a is a property of the neighbourhood of a, where " neighbourhood " is used in the sense defined in Chapter XI. during the discussion of the continuity of functions. The value of the function f(h) at a is /(a) ; but the limit is distinct in idea from the value, and may be different from it, and may exist when the value has not been defined. We shall also use the term " standard of approximation " in the sense in which it is defined in Chapter XI. In fact, in the definition of " continuity " given towards the end of that chapter we have practically defined a limit. The definition of a limit is : A function /(#) has the limit I at a value a of its argument a?, when in the neighbour- hood of a its values approximate to I within every standard of approximation. Compare this definition with that already given for continuity, namely : 230 INTRODUCTION TO MATHEMATICS A function f(x) is continuous at a value a of its argument, when in the neighbourhood of a its values approximate to its value at a within every standard of approximation. It is at once evident that a function is con- tinuous at a when (i) it possesses a limit at a, and (ii) that limit is equal to its value at a. Thus the illustrations of continuity which have been given at the end of Chapter XI. are illustrations of the idea of a limit, namely, they were all directed to proving that /(a) was the limit of /(#) at a for the functions considered and the value of a considered. It is really more instructive to consider the limit at a point where a function is not con- tinuous. For example, consider the function of which the graph is given in fig. 20 of Chap- ter XI. This function /(#) is denned to have the value 1 for all values of the argument except the integers 0, 1, 2, 3, etc., and for these integral values it has the value 0. Now let us think of its limit when x=3. We notice that in the definition of the limit the value of the function at a (in this case, a =3) is ex- cluded. But, excluding /(3), the values of /(#), when on lies within any interval which (i) contains 3 not as an end-point, and (ii) does not extend so far as 2 and 4, are all equal to 1 ; and hence these values approxi- mate to 1 within every standard of approxi- mation. Hence 1 is the limit of /(#) at the DIFFERENTIAL CALCULUS 231 value 3 of the argument x, but by definition /(3)=0. This is an instance of a function which possesses both a value and a limit at the value 3 of the argument, but the value is not equal to the limit. At the end of Chapter XI. the function x 2 was considered at the value 2 of the argument. Its value at 2 is 2 2 , i.e. 4, and it was proved that its limit is also 4. Thus here we have a function with a value and a limit which are equal. Finally we come to the case which is essen- tially important for our purposes, namely, to a function which possesses a limit, but no defined value at a certain value of its argu- ment. We need not go far to look for 2di such a function, -- will serve our purpose. x Now in any mathematical book, we might 2x find the equation, - =2, written without x hesitation or comment. But there is a diffi- culty in this ; for when x is zero, = - ; and x - has no defined meaning. Thus the value 2# of the function - - at xQ has no defined x 232 INTRODUCTION TO MATHEMATICS meaning. But for every other value of x, O/V| the value of the function is 2. Thus the x 2/p limit of at xQ is 2, and it has no value x x 2 at x=0. Similarly the limit of at xa is a whatever a may be, so that the limit of a? 2 a? 2 at 0=0 is 0. But the value of at x=0 x x takes the form -, which has no denned #2 meaning. Thus the function - - has a limit x but no value at 0. We now come back to the problem from which we started this discussion on the nature of a limit. How are we going to define the rate of increase of the function x 2 at any value x of its argument. Our answer is that this rate of increase is the limit of the func- (x -\-Tl\Z _ /r;2 tion v ^ ' at the value zero for its argument h. (Note that x is here a " con- stant.") Let us see how this answer works DIFFERENTIAL CALCULUS 233 in the light of our definition of a limit. We have (a?+A) 2 -a 2 _2/kc+A 2 _h(2x+h) h h h Now in finding the limit of ^ , ' at the value of the argument h, the value (if any) of the function at h=Q is excluded. But for all values of h, except A=0, we can divide through by h. Thus the limit of at h=0 is the same as that of 2x+h at h=0. Now, whatever standard of approximation k we choose to take, by considering the interval from \k to +\k we see that, for values of h which fall within it, 2x+h differs from 2x by less than \k, that is by less than k. This is true for any standard k. Hence in the neigh- bourhood of the value for /, 2x-\-h approxi- mates to 2x within every standard of approxi- mation, and therefore 2x is the limit of 2x+h at h=0. Hence by what has been said above 2x is the limit of ; - at the value n for h. It follows, therefore, that 2x is what we have called the rate of increase of x 2 at the value x of the argument. Thus this method conducts us to the same rate of in- 284 INTRODUCTION TO MATHEMATICS crease for x 2 as did the Leibnizian way of making h grow " infinitely small." The more abstract terms " differential co- efficient," or " derived function," are gener- ally used for what we have hitherto called the " rate of increase " of a function. The general definition is as follows : the differ- ential coefficient of the function f(x) is the limit, if it exist, of the function of the argument h at the value of its argu- ment. How have we, by this definition and the subsidiary definition of a limit, really managed to avoid the notion of " infinitely small num- bers " which so worried our mathematical forefathers ? For them the difficulty arose because on the one hand they had to use an interval x to x+h over which to calculate the average increase, and, on the other hand, they finally wanted to put h=0. The result was they seemed to be landed into the notion of an existent interval of zero size. Now how do we avoid this difficulty ? In this way we use the notion that corresponding to any standard of approximation, some in- terval with such and such properties can be found. The difference is that we have grasped the importance of the notion of " the variable," and they had not done so. Thus, DIFFERENTIAL CALCULUS 235 at the end of our exposition of the essential notions of mathematical analysis, we are led back to the ideas with which in Chapter II. we commenced our enquiry that in mathe- matics the fundamentally important ideas are those of " some things " and ** any things.'? CHAPTER XVI GEOMETRY GEOMETRY, like the rest of mathematics, is abstract. In it the properties of the shapes and relative positions of things are studied. But we do not need to consider who is observ- ing the things, or whether he becomes ac- quainted with them by sight or touch or hearing. In short, we ignore all particular sensations. Furthermore, particular things such as the Houses of Parliament, or the terrestrial globe are ignored. Every pro- position refers to any things with such and such geometrical properties. Of course it helps our imagination to look at particular examples of spheres and cones and triangles and squares. But the propositions do not merely apply to the actual figures printed in the book, but to any such figures. Thus geometry, like algebra, is dominated by the ideas of " any " and " some " things. Also, in the same way it studies the inter- relations of sets of things. For example, con- sider any two triangles ABC and DEF. 236 GEOMETRY 237 What relations must exist between some of the parts of these triangles, in order that the triangles may be in all respects equal ? This is one of the first investigations undertaken in all elementary geometries. It is a study a c E f Fig. 33. of a certain set of possible correlations be- tween the two triangles. The answer is that the triangles are in all respects equal, if : Either, (a) Two sides of the one and the in- cluded angle are respectively equal to two sides of the other and the included angle : Or, (b) Two angles of the one and the side joining them are respectively equal to two angles of the other and the side joining them : Or, (c) Three sides of the one are respect- tively equal to three sides of the other. This answer at once suggests a further en- quiry. What is the nature of the correlation between the triangles, when the three angles of the one are respectively equal to the three angles of the other ? This further investiga- tion leads us on to the whole theory of simi- 238 INTRODUCTION TO MATHEMATICS larity (cf. Chapter XIII.), which is another type of correlation. Again, to take another example, consider the internal structure of the triangle ABC. Its sides and angles are inter-related the greater angle is opposite to the greater side, and the base angles of an isosceles triangle are equal. If we proceed to trigonometry this correlation receives a more exact deter- mination in the familiar shape sin A sin B sin C a? = &2_j_ C 2 _ 2bccosA, with two similar formulae. Also there is the still simpler correlation between the angles of the triangle, namely, that their sum is equal to two right angles ; and between the three sides, namely, that the sum of the lengths of any two is greater than the length of the third Thus the true method to study geometry is to think of interesting simple figures, such as the triangle, the parallelogram, and the circle, and to investigate the correlations between their various parts. The geometer has in his mind not a detached proposition, but a figure with its various parts mutually inter-depend- ent. Just as in algebra, he generalizes the triangle into the polygon, and the side into GEOMETRY 239 the conic section. Or, pursuing a converse route, he classifies triangles according as they are equilateral, isosceles, or scalene, and polygons according to their number of sides, and conic sections according as they are hy- perbolas, ellipses, or parabolas. The preceding examples illustrate how the fundamental ideas of geometry are exactly the same as those of algebra ; except that algebra deals with numbers and geometry with lines, angles, areas, and other geo- metrical entities. This fundamental identity is one of the reasons why so many geometrical truths can be put into an algebraic dress. Thus if A, B, and C are the numbers of degrees respectively in the angles of the triangle ABC, the correlation between the angles is repre- sented by the equation and if a, b, c are the number of feet respectively in the three sides, the correlation between the sides is represented by a <(6+c, b <(c-f a, c </*+&. Also the trigonometrical formulae quoted above are other examples of the same fact. Thus the notion of the variable and the correlation of variables is just as essential in geometry as it is in algebra. But the parallelism between geometry and algebra can be pushed still further, owing to the fact that lengths, areas, volumes, and 240 INTRODUCTION TO MATHEMATICS angles are all measurable ; so that, for exam- ple, the size of any length can be determined by the number (not necessarily integral) of times which it contains some arbitrarily known unit, and similarly for areas, volumes, and angles. The trigonometrical formulae, given above, are examples of this fact. But it re- ceives its crowning application in analytical geometry. This great subject is often mis- named as Analytical Conic Sections, thereby fixing attention on merely one of its sub- divisions. It is as though the great science of Anthropology were named the Study of Noses, owing to the fact that noses are a prominent part of the human body. Though the mathematical procedures in geometry and algebra are in essence identical and intertwined in their development, there is necessarily a fundamental distinction be- tween the properties of space and the proper- ties of number in fact all the essential differ- ence between space and number. The " spaci- ness " of space and the " numerosity " of number are essentially different things, and must be directly apprehended. None of the applications of algebra to geometry or of geometry to algebra go any step on the road to obliterate this vital distinction. One very marked difference between space and number is that the former seems to be so much less abstract and fundamental than the GEOMETRY 241 latter. The number of the archangels can be counted just because they are things. When we once know that their names are Raphael, Gabriel, and Michael, and that these distinct names represent distinct beings, we know with- out further question that there are three of them. All the subtleties in the world about the nature of angelic existences cannot alter this fact, granting the premisses. But we are still quite in the dark as to their relation to space. Do they exist in space at all ? Perhaps it is equally nonsense to say that they are here, or there, or anywhere, or everywhere. Their existence may simply have no relation to localities in space. According- ly, while numbers must apply to all things, space need not do so. The perception of the locality of things would appear to accompany, or be involved in many, or all, of our sensations. It is in- dependent of any particular sensation in the sense that it accompanies many sensations. But it is a special peculiarity of the things which we apprehend by our sensations. The direct apprehension of what we mean by the positions of things in respect to each other is a thing sui generis, just as are the appre- hensions of sounds, colours, tastes, and smells. At first sight therefore it would appear that mathematics, in so far as it includes geometry in its scope, is not abstract in the sense in 242 INTRODUCTION TO MATHEMATICS which abstractness is ascribed to it in Chapter I. This, however, is a mistake ; the truth being that the " spaciness " of space does not enter into our geometrical reasoning at all. It enters into the geometrical intuitions of mathematicians in ways personal and peculiar to each individual. But what enter into the reasoning are merely certain properties of things in space, or of things forming space, which properties are completely abstract in the sense in which abstract was denned in Chapter I.; these properties do not involve any peculiar space-apprehension or space- intuition or space-sensation. They are on exactly the same basis as the mathematical properties of number. Thus the space-intui- tion which is so essential an aid to the study of geometry is logically irrelevant : it does not enter into the premisses when they are properly stated, nor into any step of the rea- soning. It has the practical importance of an example, which is essential for the stimulation of our thoughts. Examples are equally neces- sary to stimulate our thoughts on number. When we think of " two " and " three " we see strokes in a row, or balls in a heap, or some other physical aggregation of particular things. The peculiarity of geometry is the fixity and overwhelming importance of the one particular example which occurs to our GEOMETRY 243 minds. The abstract logical form of the propositions when fully stated is, " If any collections of things have such and such abstract properties, they also have such and such other abstract properties.'-' But what appears before the mind's eye is a collection of points, lines, surfaces, and volumes in the space : this example inevitably appears, and is the sole example which lends to the propo- sition its interest. However, for all its over- whelming importance, it is but an example. Geometry, viewed as a mathematical science, is a division of the more general science of order. It may be called the science of dimen- sional order ; the qualification " dimensional " has been introduced because the limitations, which reduce it to only a part of the general science of order, are such as to produce the regular relations of straight lines to planes, and of planes to the whole of space. It is easy to understand the practical im- portance of space in the formation of the scientific conception of an external physical world. On the one hand our space-percep- tions are intertwined in our various sensations and connect them together. We normally judge that we touch an object in the same place as we see it ; and even in abnormal cases we touch it in the same space as we see it, and this is the real fundamental fact which ties together our various sensations. Accord- 244 INTRODUCTION TO MATHEMATICS ingly, the space perceptions are in a sense the common part of our sensations. Again it happens that the abstract properties of space form a large part of whatever is of spatial interest. It is not too much to say that to every property of space there corresponds an abstract mathematical statement. To take the most unfavourable instance, a curve may have a special beauty of shape : but to this shape there will correspond some abstract mathematical properties which go with this shape and no others. Thus to sum up : (1) the properties of space which are investigated in geometry, like those of number, are properties belonging to things as things, and without special reference to any particular mode of apprehension : (2) Space-perception accompanies our sensations, perhaps all of them, certainly many ; but it does not seem to be a necessary quality of things that they should all exist in one space or in any space. CHAPTER XVII QUANTITY IN the previous chapter we pointed out that lengths are measurable in terms of some unit length, areas in term of a unit area, and volumes in terms of a unit volume. When we have a set of things such as lengths which are measurable in terms of any one of them, we say that they are quantities of the same kind. Thus lengths are quantities of the same kind, so are areas, and so are volumes. But an area is not a quantity of the same kind as a length, nor is it of the same kind as a volume. Let us think a little more on what is meant by being measurable, taking lengths as an example. Lengths are measured by the foot-rule. By transporting the foot-rule from place to place we judge of the equality of lengths. Again, three adjacent lengths, each of one foot, form one whole length of three feet. Thus to measure lengths we have to determine the equality of lengths and the addition of lengths. When some test has been applied, such as the transporting of a foot-rule, we say that the lengths are equal ; and when some process 245 has been applied, so as to secure lengths being contiguous and not overlapping, we say that the lengths have been added to form one whole length. But we cannot arbitrarily take any test as the test of equality and any process as the process of addition. The re- sults of operations of addition and of judg- ments of equality must be in accordance with certain preconceived conditions. For exam- ple, the addition of two greater lengths must yield a length greater than that yielded by the addition of two smaller lengths. These preconceived conditions when accurately for- mulated may be called axioms of quantity. The only question as to their truth or falsehood which can arise is whether, when the axioms are satisfied, we necessarily get what ordinary people call quantities. If we do not, then the name " axioms of quantity " is ill-judged that is all. These axioms of quantity are entirely ab- stract, just as are the mathematical properties of space. They are the same for all quantities, and they presuppose no special mode of per- ception. The ideas associated with the notion of quantity are the means by which a con- tinuum like a line, an area, or a volume can be split up into definite parts. Then these parts are counted ; so that numbers can be used to determine the exact properties of a continuous whole. QUANTITY 247 Our perception of the flow of time and of the succession of events is a chief example of the application of these ideas of quantity. We measure time (as has been said in con- sidering periodicity) by the repetition of similar events the burning of successive inches of a uniform candle, the rotation of the earth relatively to the fixed stars, the rotation of the hands of a clock are all ex- amples of such repetitions. Events of these types take the place of the foot-rule in rela- tion to lengths. It is not necessary to assume that events of any one of these types are exactly equal in duration at each recurrence. What is necessary is that a rule should be known which will enable us to express the relative durations of, say, two examples of some type. For example, we may if we like suppose that the rate of the earth's rotation is decreasing, so that each day is longer than the preceding by some minute fraction of a second. Such a rule enables us to compare the length of any day with that of any other day. But what is essential is that one series of repetitions, such as successive days, should be taken as the standard series ; and, if the various events of that series are not taken as of equal duration, that a rule should be stated which regulates the duration to be assigned to each day in terms of the duration of any other day. 248 INTRODUCTION TO MATHEMATICS What then are the requisites which such a rule ought to have ? In the first place it should lead to the assignment of nearly equal durations to events which common sense judges to possess equal durations. A rule which made days of violently different lengths, and which made the speeds of apparently similar operations vary utterly out of pro- portion to the apparent minuteness of their differences, would never do. Hence the first requisite is general agreement with common sense. But this is not sufficient absolutely to determine the rule, for common sense is a rough observer and very easily satisfied. The next requisite is that minute adjustments of the rule should be so made as- to allow of the simplest possible statements of the laws of nature. For example, astronomers tell us that the earth's rotation is slowing down, so that each day gains in length by some incon- ceivably minute fraction of a second. Their only reason for their assertion (as stated more fully in the discussion of periodicity) is that without it they would have to abandon the Newtonian laws of motion. In order to keep the laws of motion simple, they alter the measure of time. This is a perfectly legiti- mate procedure so long as it is thoroughly understood. What has been said above about the ab- stract nature of the mathematical properties QUANTITY of space applies with appropriate verbal changes to the mathematical properties of time. A sense of the flux of time accompanies all our sensations and perceptions, and prac- tically all that interests us in regard to time can be paralleled by the abstract mathe- matical properties which we ascribe to it. Conversely what has been said about the two requisites for the rule by which we determine the length of the day, also applies to the rule for determining the length of a yard measure namely, the yard measure appears to retain the same length as it moves about. Accord- ingly, any rule must bring out that, apart from minute changes, it does remain of in- variable length; Again, the second requisite is this, a definite rule for minute changes shall be stated which allows of the simplest expression of the laws of nature. For ex- ample, in accordance with the second re- quisite the yard measures are supposed to expand and contract with changes of tem- perature according to the substances which they are made of. Apart from the facts that our sensations are accompanied with perceptions of locality and of duration, and that lines, areas, volumes, and durations, are each in their way quanti- ties, the theory of numbers would be of very subordinate use in the exploration of the laws of the Universe,- As it is, physical science reposes on the main ideas of number, quan- tity, space, and time. The mathematical sciences associated with them do not form the whole of mathematics, but they are the substratum of mathematical physics as at present existing. NOTES A (p. 60). In reading these equations it must be noted that a bracket is used in mathematical symbolism to mean that the operations within it are to be performed first. Thus (l + 3)+2 directs us first to add 3 to 1, and then to add 2 to the result; and l + (3+2) directs us first to add 2 to 3, and then to add the result to 1. Again a numerical example of equation (5) is 2x(3+4)=(2x3) + (2x4). We perform first the operations in brackets and obtain 2x7=6+8 which is obviously true. SP B (p. 136). This fundamental ratio -p^ is called the eccentricity of the curve. The shape of the curve, as distinct from its scale or size, depends upon the value of its eccentricity. Thus it is wrong to think of ellipses in general or of hyperbolas in general as having in either case one definite shape. Ellipses with different eccen- tricities have different shapes, and their sizes depend upon the lengths of their major axes. An ellipse with small eccentricity is very nearly a circle, and an ellipse of eccentricity only slightly less than unity is a long flat oval. All parabolas have the same eccentricity and are therefore of the same shape, though they can be drawn to different scales. BIBLIOGRAPHY 251 C (p. 204). If a series with all its terms positive is convergent, the modified series found by making some terms positive and some negative according to any definite rule is also convergent. Each one of the set of series thus found, including the original series, is called " absolutely convergent." But it is possible for a series with terms partly positive and partly negative to be convergent, although the corresponding series with all its terms positive is divergent. For example, the series 1-i+i-i+etc. is convergent though we have just proved that l+i-H+i+ etc. is divergent. Such convergent series, which are not absolutely convergent, are much more difficult to deal with than absolutely convergent series. BIBLIOGRAPHY NOTE ON THE STUDY OF MATHEMATICS THE difficulty that beginners find in the study of this science is due to the large amount of technical detail which has been allowed to accumulate in the elementary text- books, obscuring the important ideas. The first subjects of study, apart from a knowledge of arithmetic which is presupposed, must be elementary geometry and elementary algebra. The courses in both subjects should be short, giving only the necessary ideas ; the algebra should be studied graphically, so that in practice the ideas of elementary coordinate geometry are also being assimilated. The next pair of subjects should be elementary trigonometry and the coordinate geometry of the straight line and circle. The latter subject is a short one ; for it really merges into the algebra. The student is then prepared to enter upon conic sections, a very short course of geometrical conic sections and a longer one of analytical conies. But in all these courses great care should be taken not to overload the mind with more 252 BIBLIOGRAPHY detail than is necessary for the exemplification of the fundamental ideas. The differential calculus and afterwards the integral calculus now remain to be attacked on the same system. A good teacher will already have illustrated them by the consideration of special cases in the course on algebra and coordinate geometry. Some short book on three- dimensional geometry must be also read. This elementary course of mathematics is sufficient for some types of professional career. It is also the necessary preliminary for any one wishing to study the subject for its intrinsic interest. He is now prepared to commence on a more extended course. He must not, however, hope to be able to master it as a whole. The science has grown to such vast proportions that probably no living mathe- matician can claim to have achieved this. Passing to the serious treatises on the subject to be read after this preliminary course, the following may be men- tioned : Cremona's Pure Geometry (English Translation, Clarendon Press, Oxford), Hobson's Treatise on Trigono- metry, ChrystaPs Treatise on Algebra (2 volumes), Salmon's Gonic Sections, Lamb's Differential Calculus, and some book on Differential Equations. The student will probably not desire to direct equal attention to all these subjects, but will study one or more of them, according as his interest dictates. He will then be prepared to select more ad- vanced works for himself, and to plunge into the higher parts of the subject. If his interest lies in analysis, he should now master an elementary treatise on the theory of Functions of the Complex Variable ; if he prefers to specialize in Geometry, he must now proceed to the standard treatises on the Analytical Geometry of three dimensions. But at this stage of his career in learning he will not require the advice of this note. I have deliberately refrained from mentioning any elementary works. They are very numerous, and of various merits, but none of such outstanding superiority as to require special mention by name to the exclusion of all the others. INDEX Abel, 156 Abscissa, 95 Absolute Convergence, 251 Abstract Nature of Geo- metry, 242 et aeqq. Abstractness (defined), 9, 13 Adams, 220 Addition Theorem, 212 Ahmes, 71 Alexander the Great, 128, 129 Algebra.Fundamental Laws of, 60 Ampere, 34 3 Analytical Conic Sections, 240 Apollonius of Perga, 131, 134 j Approximation, 197 et aeqq. Arabic Notation, 58 et aeqq. Archimedes, 37 et aeqq. , Argument of a Function, 146 i Aristotle, 30, 42, 128 Astronomy, 137, 173, 174 Axes, 125 Axioms of Quantity, 246 et aeqq. Bacon, 156 Ball, W. W. R., 58 Beaconsfield, Lord, 41 Berkeley, Bishop, 226 Bhaskara, 58 Cantor, Georg, 79 Circle, 120, 130, 180 et aeqq. Circular Cylinder, 143 Clerk Maxwell, 34, 35 Columbus, 122 Compact Series, 76 Complex Quantities, 109 .Conic Sections, 128 et aeqq. Constants, 69, 117 Continuous Functions, 150 et aeqq. ; 162 (defined) Convergence, Absolute, 251 Convergent, 203 et aeqq. Coordinate Geometry, 112 et aeqq. Coordinates, 95 Copernicus, 45, 137 Coaine, 182 et aeqq. Coulomb, 33 Cross Ratio, 140 Darwin, 138, 220 Derived Function, 234 Descartes, 95, 113, 116, 122, 218 Differential Calculus, 217 et aeqq. Differential Coefficient, 234 Directrix, 135 Discontinuous Functions, 150 et aeqq. Distance, 30 Divergent, 203 et aeqq. 253 254 INDEX Dynamical Explanation, 13, 14, 47 et segq. Dynamics, 30, 43 et seqq. Eccentricity, 250 Electric Current, 33 Electricity, 32 et seqq. Electromagnetism, 31 et seqq. Ellipse, 45, 120, 130 et seqq. Euclid, 114 Exponential Series, 211 et seqq. Faraday, 34 Format, 218 Fluxions, 219 Focus, 120, 135 Force, 30 Form, Algebraic, 66 et seqq., 82, 117 Fourier's Theorem, 191 Fractions, 71 et seqq. Franklin, 32, 122 Function, 144 et seqq. Galileo, 30, 42 et seqq., 122 Galvani, 33 Generality in Mathematics, 82 Geometrical Series, 206 et seqq. Geometry, 36, 236 et seqq. Gilbert, Dr., 32 Graphs, 148 et seqq. Gravitation, 29, 139 Halley, 139 Harmonic Analysis, 192 Harriot, Thomas, 66 Here, 35 Hiero, 38 Hipparchus, 173 Hyperbola, 131 et seqq. Imaginary Numbers, 87 et seqq. Imaginary Quantities, 109 Incommensurable Ratios, 72 et seqq. Infinitely Small Quantities, 226 et seqq. Integral Calculus, 222 Interval, 158 et seqq. Kepler, 45, 46, 137, 138 Kepler's Laws, 138 Laputa, 10 Laws of Motion, 167 et seqq., 248 Leibniz, 16, 218 et seqq. Leonardo da Vinci, 42 Leverrier, 220 Light, 35 Limit of a Function, 227 et seqq. Limit of a Series, 199 et seqq. Limits, 77 Locus, 121 etseqq., 141 Macaulay, 156 Malthus, 138 Marcellus, 37 Mass, 30 Mechanics, 46 Menaechmus, 128, 129 Motion, First Law of, 43 INDEX 255 Neighbourhood, 159 et aeqq. Newton, 10, 16, 30, 34, 37, 38, 43, 46, 139, 218 et aeqq. Non-Uniform Convergence, 208 et aeqq. Normal Error, Curva of, 214 Oersted, 34 Order, 194 et aeqq. Order, Type of, 75 et aeqq., 196 Ordered Couples, 93 et aeqq. Ordinate, 95 Origin, 95, 126 Pappus, 135, 136 Parabola, 131 et aeqq. Parallelogram Law, 61 et aeqq., 99, 126 Parameters, 69, 117 Pencils, 140 Period, 170, 189 etaeqq. Periodicity, 164 et seqq., 188, 216 Pitt, William, 194 Pizarro, 122 Plutarch, 37 Positive and Negative Numbers, 83 et aeqq. Projective Geometry, 139 Ptolemy, 137, 173 Pythagoras, 18 Quantity, 245 et aeqq. Rate of Increase of Func- tions, 220 et aeqq. Ratio, 72 et aeqq. Real Numbers, 73 et aeqq. Rectangle, 57 Relations between Vari- ables, 18 et aeqq. Resonance, 170, 171 Rosebery, Lord, 194 Scale of a Map, 178 Seidel, 210 Series, 74 et aeqq., 194 et aeqq. Shelley (quotation from), 217 Similarity, 177 et seqq., 237 Sine, 182 et aeqq. Specific Gravity, 41 Squaring the Circle, 187 Standard of Approxima- tion, 159 et aeqq., 201 et aeqq., 229 etseqq. Steps, 79 et aeqq., 96 Stifel, 85 Stokes, Sir George, 210 Sum to Infinity, 201 et aeqq. Surveys, 176 et aeqq. Swift, 10 Tangents, 221, 222 Taylor's Theorem, 166, 157 Time, 166 et aeqq., 247 et aeqq. Transportation, Vector of, 54 et aeqq. Triangle, 176 et aeqq., 237 Triangulation, 177 Trigonometry, 173 et aeqq. Uniform Convergence, 208 et aeqq. Unknown, The, 17, 23 256 INDEX Value of a Function, 146 Variable, The, 18, 24, 49, 82, 234, 239 Variable Function, 147 Vectors, 51 et eeqq., 85, 96 Vertex, 134 Volta, 33 Wallace, 220 Weierstrass, 156, 226, 228 Zero, 63 et seqq., 108 Printed fry Hazell, Walton & Viney, Ld., London and Ayktbwy. The Home University Library of Modern Knowledge jj Comprehensive Series of New and Specially Written (Books EDITORS : PROF. GILBERT MURRAY, D.Litt., LL.D., F.B.A. HERBERT FISHER, M.A., F.B.A. PKOF. J. ARTHUR THOMSON, M.A. PROF. WM. T. BREWSTER, M.A. The Home University Library " Is without the slightest doubt the pioneer in supplying serious literature for a large section of the public who are interested in the liberal educa- tion of the State." The Daily Mail. " It is a thing very favourable to the real success of The Home University Library that its volumes do not merely attempt to feed ignorance with knowledge. The authors noticeably realise that the simple willing appetite of sharp-set ignorance is not specially common nowadays ; what is far more common is a hunger which has been partially but injudiciously filled, with more or less serious results of indigestion. The food supplied is therefore frequently medicinal as well as nutritious ; and this is certainly what the time requires. " Manchester Guardian. "Each volume represents a three-hours' traffic with the talking-power of a good brain, operating with the ease and interesting freedom of a specialist dealing with his own subject. ... A series which promises to perform a real social service." The Times. "We can think of no series now being issued which better deserves support." The Observer. We think if they were given as prizes in place of the more costly di: id prol series they might well take : ispensed on prize days, the pupils would rofit. If the publishers want a motto for the rubbish that is wont to be find more pleasure and series they might well take : ' Infinite riches in a little room.'" Irish Journal of Education, " The scheme was successful at the start because it met a want among earnest readers; but its wider and* sustained success, surely, comes from the fact that it has to a large extent created and certainly refined the taste by which it is appreciated." Daily Chronicle. " Here is the world's learning in little, and none too poor t< house-room!" Daily Telegraph. to give it ]/- net in cloth 256 Pages 2/6 net in leather History and (geography 3. THE FRENCH REVOLUTION By HILAIKE BELLOC, M.A. (With Maps.) "It is coloured with all the militancy of the author's temperament." Daily News. 4. HISTORY OF WAR AND PEACE By G. H. FERRIS. The Rt. Hon. JAMES BRYCE writes: " I have read it with much interest and pleasure, admiring the skill with which you have managed to compress so many facts and views into so small a volume." 8. POLAR EXPLORATION By Dr W. S. BRUCE, F.R.S.E., Leader of the "Scotia" Expedition. (With Maps.) "A very freshly written and interesting narrative." The Times. "A fascinating book." Portsmouth Times. 12. THE OPENING-UP OF AFRICA By Sir H. H. JOHNSTON. G.C.M.G., K.C.B., D.Sc., F.Z.S. (With Maps.) " The Home University Library is much enriched by this excellent work." Daily Mail. 13. MEDIAEVAL EUROPE By H. W. C. DAVIS, M.A. (With Maps.) "One more illustration of the fact that it takes a complete master of the subject to write briefly upon it." Manchester Guardian. 14. THE PAPACY &* MODERN TIMES (1303-1870) By WILLIAM BARRY, D.D. "Dr Barry has a wide range of knowledge and an artist's power of selection." Manchester Guardian. 23. HISTORY OF OUR TIME, 1885-1911 By G. P. GOOCH, M.A. " Mr Gooch contrives to breathe vitality into his story, and to give us the flesh as well as the bones of recent happenings." Observer. 25. THE CIVILISATION OF CHINA By H. A. GILES, LL.D., Professor of Chinese in the University of Cambridge. "In all the mass of facts, Professor Giles never becomes dull. He is always ready with a ghost story or a street adventure for the reader's recreation." Spectator. 29. THE DA WN OF HISTORY By J.L.MYRES, M. A., F.S. A., Wykeham Professor of Ancient History, Oxford. "There is not a page in it that is not suggestive." Manchester Guardian. 33. THE HISTORY OF ENGLAND: A Study in Political Evolution. By Prof. A. F. POLLARD, M.A. With a Chronological Table. " It takes its place at once among the authoritative works on English history." Observer. 34. CANADA By A. G. BRADLEY. " Who knows Canada, better than Mr A. G. Bradley? " Daily Chronicle. "The volume makes an immediate appeal to the man who wants to know something vivid and true about Canada." Canadian Gazette. 37. PEOPLES 6* PROBLEMS OF INDIA By Sir T. W. HOLDERNESS, K.C.S.I., Secretary of the Revenue, Statistics, ! and Commerce Department of the India Office. "Just the book which news- paper readers require to-day, and a marvel of comprehensiveness." Pall \ Mall Gazette. 42. ROME By W. WARDE FOWLHR, M.A. " A masterly sketch of Roman character and of what it did for the world." The Spectator. "It has all the lucidity and charm of presentation we expect from this writer." Manchester Guardian. 48. THE AMERICAN CIVIL WAR By F. L. PAXSON, Professor of American History, Wisconsin University. (With Maps.) "A stirring study." The Guardian. 51. WARFARE IN BRITAIN By HILAIRE BELLOC, M.A. An account of how and where great battles of the past were fought on British soil, the roads and physical conditions determining the island's strategy, the castles, walled towns, etc. 55. MASTER MARINERS By J. R. SPEARS. The romance of the sea, the great voyages of discovery, naval battles, the heroism of the sailor, and the development of the ship, from ancient times to to-day. IN PREPARATION ANCIENT GREECE. By Prof. GILBERT MURRAY, D.Litt., LL.D., F.B.A ANCIENT EGYPT. By F. LL. GRIFFITH, M.A. THE ANCIENT EAST. By D. G. HOGARTH, M.A., F.B.A. A SHORT h'ISTOR YOFEUROPE. By HERBERT FISHER, M. A., F.B.A. PREHISTORIC BRITAIN. By ROBERT MUNRO, M.A., M.D., LL.D. THE BYZANTINE EMPIRE. By NORMAN H. BAVNES. THE REFORM A TION. By Principal LINDSAY, LL.D. NAPOLEON. By HERBERT FISHER, M.A., F.B.A. A SHORT HISTORY OF RUSSIA. By Prof. MILYOUKOV. MODERN TURKEY. By D. G. HOGARTH, M.A. FRANCE OF TO-DAY. By ALBERT THOMAS. GERMANY OF TO-DA Y. By CHARLES TOWER. THE NAVY AND SEA POWER. By DAVID HANNAY. HISTORY OF SCOTLAND. By R. S. RAIT, M.A. SOUTH AMERICA. By Prof. W. R. SHEPHERD. LONDON. By Sir LAURENCE GOMME, F.S.A. HISTORY AND LITERATURE OF SPAIN. By J. FITZMAURICE- KELLY, F.B.A., Litt.D. Literature and 2. SHAKESPEARE By JOHN MASEFIELD. " The book is a joy. We have had half-a-dozen more learned books on Shakespeare in the last few years, but not one so wise." Manchester Guardian. 27. ENGLISH LITERATURE: MODERN By G. H. MAIR, M.A. " Altogether a fresh and individual book." Olstrver. 35. LANDMARKS IN FRENCH LITERATURE By G. L. STRACHEY. " Mr Strachey is to be congratulated on his courage and success. It is difficult to imagine how a better account of French Literature could be given in 250 small pages than he has given here." The Times. 39- ARCHITECTURE By Prof. W. R. LETHABY. (Over forty Illustrations.) " Popular guide-books to architecture are, as a rule, not worth ranch. This volume is a welcome excep- tion." Building News. " Delightfully bright reading." Christian World. 43. ENGLISH LITERATURE: MEDIAEVAL By Prof. W. P. KER, M.A. "Prof. Ker has long proved his worth as one of the soundest scholars in English we have, and he is the very man to put an outline of English Mediaeval Literature before the uninstructed public. His knowledge and taste are unimpeachable, and his style is effective, simple, yet never dry." The Athemeum. 45. THE ENGLISH LANGUAGE By L. PEARSALL SMI-TH, M.A. "A wholly fascinating study of the different streams that went to the making of the great river of the English speech." Daily News. 52. GREAT WRITERS OF AMERICA By Prof. J. EKSKINE and Prof. W. P. TRENT. A popular sketch by two foremost authorities. IN PREPARATION ANCIENT ART AND RITUAL. By Miss JANE HARRISON, LL.D., D.Litt. GREEK LITERA TURE. By Prof. GILBERT MURRAY, D.Litt. LA TIN LITER A TURE. By Prof. J. S. PHILLIMORE. CHA UCER AND HIS TIME. By Miss G. E. HADOW. THE RENAISSANCE. By Mrs R. A. TAYLOR. ITALIAN A RTOF THE RENAISSANCE. By ROGER E. FRY, M.A. THE ART OF PAINTING. By Sir FREUERICK WEDMORE. DR JOHNSON AND HIS CIRCLE. By JOHN BAILEY, M.A. THE VIC IORIAN AGE. By G. K- CHESTERTON. ENGLISH COMPOSITION. By Prof. WM. T. BREWSTER. GREA T WRITERS OF RUSSIA. By C. T. HAGBERG WRIGHT, LL.D. THE LITERATURE OF GERMANY. By Prof. J. G. ROBERTSON, M.A., Ph.D. SCANDINAVIAN HISTORY AND LITERATURE. By T. C. SNOW, M.A. Science 7. MODERN GEOGRAPHY By Dr MARION NEWBIGIN. (Illustrated.) "Geography, again : what a dull, tedious study that was wont to be I . . . But Miss Marion Newbigin invests its dry bones with the flesh and blood of romantic interest, taking stock of geography as a fairy-book of science." Daily Telegraph. 9. THE EVOLUTION OF PLANTS By Dr D. H. SCOTT, M.A., F.R.S., late Hon. Keeper of the Jodrell Laboratory, Kew. (Fully illustrated.) "The information which the book provides is as trustworthy as first-band knowledge can make it. ... Dr Scott's candid and familiar style makes the difficult subject both fascinating and easy." Gardeners' Chronicle. 17. HEALTH AND DISEASE By W. LESLIE MACKKNZIE, M.D., Local Government Board, Edinburgh. "The science of public health administration has had no abler or more attractive exponent than Dr Mackenzie. He adds to a thorough grasp of the problems an illuminating style, and an arresting manner of treating a subject often dull and sometimes unsavoury." Economist. 1 8. INTRODUCTION TO MATHEMATICS ' By A. N. WHITEHEAD, Sc.D., F.R.S. (With Diagrams.) "MrWhitehead has discharged with conspicuous success the task he is so exceptionally qualified I to undertake. For he is one of our great authorities upon the foundations of the science, and has the breadth of view which is so requisite in presenting to the reader its aims. His exposition is clear and striking." Westminster Gazette. 19. THE ANIMAL WORLD By Professor F. W. GAMBLE, D.Sc., F.R.S. With Introduction hy Sir Oliver Lodge. (Many Illustrations.) " A delightful and instructive epitome of animal (and vegetable) life. ... A most fascinating and suggestive survey." Morning Post. 20. EVOLUTION By Professor J. ARTHUR THOMSON and Professor PATRICK GEDDES. "A many-coloured and romantic panorama, opening up, like no other book we know, a rational vision of world-development." Belfast News-Letter. 22. CRIME AND INSANITY By Dr C. A. MERCIER, F.R.C.P., F.R.C.S., Author of "Text-Book of In- sanity," etc- " Furnishes much valuable information from one occupying the highest position among medico-legal psychologists." Asylum NCVJS. 28. PSYCHICAL RESEARCH and thus what he has to say on thought-reading, hypnotism, telepathy, crystal- vision, spiritualism, divinings, and so on, will be read with avidity." Dundee 31. ASTRONOMY By A. R. HINKS, M.A., Chief Assistant, Cambridge Observatory. "Original in thought, eclectic in substance, and critical in treatment. . . . No better little book is available." School World. 32. INTRODUCTION TO SCIENCE By J. ARTHUR THOMSON, M.A., Regius Professor of Natural History, Aberdeen University. " Professor Thomson's delightful literary style is well known ; and here he discourses freshly and easily on the methods of science and its relations with philosophy, art, religion, and practical life." Aberdeen Journal, 36. By H. N. DICKSON, D.Sc. Oxon., M.A., F.R.S.E., President of the Royal Meteorological Society ; Professor of Geography in University College, Reading. (With Diagrams.) "The author has succeeded in presenting in a very lucid and agreeable manner the causes of the movement of the atmosphere and of the more stable winds." Manchester Guardian. 41. ANTHROPOLOGY By R R. MARETT, M.A., Reade "An absolutely perfect handboo fascinating and human that it bea 44. THE PRINCIPLES OF PHYSIOLOGY By Prof. J. G. McKENDRiCK, M.D. " It is a delightful and wonderfully com- prehensive handling of a subject which, while of importance to all, does not readily lend itself to untechnical explanation. . . . The little book is more than a mere repository of knowledge ; upon every page of it is stamped the impress of a creative imagination." Glasgow Herald. By R. R. MARETT, M.A., Reader in Social Anthropology in Oxford University. "An absolutely perfect handbook, so clear that a child could understand it, so fascinating and human that it beats fiction ' to a frazzle.' " Morning Leader. 46. MATTER AND ENERGY By F. SODDY, M.A., F.R.S. "A most fascinating and instructive account or the great facts of physical science, concerning which our knowledge, of later years, has made such wonderful progress." The Bookseller. 49. PSYCHOLOGY, THE STUDY OF BEHAVIOUR By Prof. W. McDouGALL, F.R.S., M.B. "A happy example of the non- technical handling of an unwieldy science, suggesting rather than dogmatising. It should whet appetites for deeper study." Christian World. 53. THE MAKING OF THE EARTH ByProf.J.W. GREGORY, F.R.S. (With 38 Maps and Figures.) The Professor of Geology at Glasgow describes the origin of the earth, the formation and changes of its surface and structure, its geological history, the first appearance of life, and its influence upon the globe. 57. THE HUMAN BODY By A. KEITH, M.D., LL,D., Conservator of Museum and Hunterian Pro- fessor, Royal College of Surgeons. (Illustrated.) The work of the dissecting- room is described, and among other subjects dealt with are : the development of the body ; malformations and monstrosities ; changes of youth and age ; sex differences, are they increasing or decreasing? race characters ; bodily features as indexes of mental character ; degeneration and regeneration ; and the genealogy and antiquity of man. 58. ELECTRICITY By GisBERT KAPP, D.Eng., M.I.E.E., M.I.C.E., Professor of Electrical Engineering in the University of Birmingham. (Illustrated.) Deals with frictional and contact electricity ; potential ; electrification by mechanical means ; the electric current ; the dynamics of electric currents ; alternating currents ; the distribution of electricity, etc. IN PREPARATION CHEMISTRY. Py Prof. R. MELDOLA, F.R.S. THE MINERAL WORLD. By Sir T. H. HOLLAND, K.C.I. E., D.Sc. PLANT LII-'E. By Prof. J. B. FARMER, F.R.S. NERVES. By Prof. D. FRASER HARRIS, M.D., D.Sc. A STUDY OF SEX. By Prof. J. A. THOMSON and Prof. PATRICK GEDDES. THE GROWTH OF EUROPE. By Prof. GRKNVILLE COLE. Philosophy and "Religion ig's :tate 15. MOHAMMEDANISM By Prof. D. S. MARGOLIOUTH, M.A., D.Litt. "This generous shilling': worth of wisdom. ... A delicate, humorous, and most responsible tractati by an illuminative professor." Daily Mail. 40. THE PROBLEMS OF PHILOSOPHY By the Hon. BERTRAND RUSSELL, F.R.S. : 'A book that the ' man in the street ' will recognise at once to be a boon. . . . Consistently lucid and non- technical throughout." Christian World. 47. BUDDHISM go. NONCONFORMITY: Its ORIGIN and PROGRESS I'.'- Principal W. B. SELBIE, M.A. "The historical part is brilliant in its :., clarity, and proportion, and in the later chapters on the present position .urns of Nonconformity Dr Selbie proves himself to be an ideal exponent of sound and moderate views." Christian World. 54. ETHICS By G. E. MOORE, M.A., Lecturer in Moral Science in Cambridge University. Discusses Utilitarianism, the Objectivity of Moral Judgments, the Test of Right and Wrong, Free Will, and Intrinsic Value. 56. THE MAKING OF THE NEW TESTAMENT By Prof. B. W. BACON, LL. LX, D.D. An authoritative summary of the results of modern critical research with regard to the origins of the New Testament, in " the formative period when conscious inspiration was still in its full glow rather than the period of collection into an official canon," showing the mingling of the two great currents of Christian thought " Pauline and 'Apostolic,' the Greek- Christian gospel about Jesus, and the Jewish-Christian gospel of Jesus, the gospel of the Spirit and the gospel of au thority." jo. MISSIONS: THEIR RISE and DEVELOPMENT By Mrs CREIGHTON. The beginning of modern missions after the Reforma- tion and their growth are traced, and an account is given of their present work, its extent and character. IN PREPARATION THE OLD TESTAMENT. By Prof. GEORGE MOORE, D.D., LL.D. BETWEEN THE OLD AND NEW TESTAMENTS. By R. H. CHARLES, D.D. COMPARATIVE RELIGION. By Prof. J. ESTLIN CARPENTER, D.Litt. A HISTOR Y of FREEDOM of THOUGHT. By Prof. J. B. BURY, LL.D. A HISTORY OF PHILOSOPHY. By CLEMKNT WKBB, M.A. Social Science . PARLIAMENT Its History, Constitution, and Practice. By Sir COURTENAY P. ILBERT. K.C.B., K.C.S.I., Clerk of the House of Commons. "The best book on the history and practice of the House of Commons since Bagehot's 'Constitution.'" Yorkshire Post. . THE STOCK EXCHANGE By F. W. HIRST, Editor of " The Economist." " To an unfinancial mind must be a revelation. . . . The book is as clear, vigorous, and sane as Bagehot's ' Lom- bard Street,' than which there is no higher compliment." Morning Leader . IRISH NATIONALITY By Mrs J. R. GREEN. " As glowing as it is learned. No book could be more timely." Daily News. "A powerful study. . . . A magnificent demonstration of the deserved vitality of the Gaelic spirit." Freeman s Journal. 3. THE SOCIALIST MOVEMENT RAMSAY MACDONALD, M.T. "Admirably adapted for the purpose of exposition." The Times. "Mr MacDonald is a very lucid exponent. . . . The volume will be of great use in dispelling illusions about the tendencies of Socialism in this country." The Nation. i. CONSERVATISM Jy Lord HUGH CECIL, M.A., M.P. "One of those great little books which seldom appear more than once in a generation." Morning Post. 1 6. THE SCIENCE OF WEALTH By J. A. HOUSON, M.A. "Mr J. A. Hobson holds an unique position among living economists. . . . The text-book produced is altogether admirable. Original, reasonable, and illuminating." The Nation. 21. LIBERALISM By L. T. HOBHOUSE, M. A., Professor of Sociology in the University of London. "A book of rare quality. . . . We have nothing but praise for the rapid and masterly summaries of the arguments from first principles which form a large part of this book." Westminster Gazette. 24. THE EVOLUTION OF INDUSTRY ByD. H. MACGREGCR, M.A., Professor of Political Economy in the University of Leeds. "A volume so dispassionate in terms may be read with profit by all interested in the present state of unrest." Aberdeen Journal. 26. AGRICULTURE By Prof. W. SOMERVILLE, F.L.S. " It makes the results of laboratory work at the University accessible to the practical farmer." Athena-urn. 30. ELEMENTS OF ENGLISH LA W By W. M. GELDART, M.A., B.C.L., Vinerian Professor of English Law at Oxford. "Contains a very clear account of the elementary principles under- lying the rules of English law ; and we can recommend it to all who wish to become acquainted with these elementary principles with a minimum of trouble." Scots Law Times. 38. THE SCHOOL An Introduction to the Study of Education. By J. J. FINDLAY, M.A., Ph.D., Professor of Education in Manchester University. <: An amazingly comprehensive volume. . . . It is a remarkable performance, distinguished in its crisp, striking phraseology as well as its inclusiveness of subject-matter." Morning Post. -59. ELEMENTS OF POLITICAL ECONOMY By S. J. CHAPMAN, M.A., Professor of Political Economy in Manchester University. A simple explanation, in the light of the latest economic thought, of the working of demand and supply ; the nature of monopoly ; money and international trade ; the relation of wages, profit, interest, and rent ; and the effects of labour combination prefaced by a short sketch of economic study since Adam Smith. IN PREPARATION THE CRIMINAL AND THE COMMUNITY. By Viscount ST. CYRES, M.A. COMMONSENSE IN LA W. By Prof. P. VINOGRADOFF, D.C.L. THE CIVIL SERVICE. By GRAHAM WALLAS, M.A. PRACTICAL IDEALISM. By MAURICE HEWLETT. NEWSPAPERS. By G. BINNEY DIBBLEE. ENGLISH VILLAGE LIFE. By E. N. BENNETT, M.A. CO -PARTNERSHIP At\D PROFIT-SHARING. By ANEURIN WILLIAMS, J.P. THE SOCIAL SETTLEMENT. By JANE ADDAMS and R. A. WOODS. GREA T INVENTIONS. By Prof. J. L. MYRES, M.A., F.S.A. TOWN PLANNING. By RAYMOND UNWIN. POLITICAL THOUGHT IN ENGLAND: From Bentham to J. S. Mill. By Prof. W. L. DAVIDSON. POLITICAL THOUGHT IN ENGLAKD: From Herbert Spencer to To-day. By ERNEST BARKER, M.A. London: WimTMS^AND~NORGATE And of all Bookshops and Bookstalls. THE LIBRARY UNIVERSITY OF CALIFORNIA Santa Barbara THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW. UCT91989 SETOSE? 2 3 1205 00394 1521 UC SOUTHERN REGIONAL LIBRARY FACILITY A 000 804 550 2